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Off-the-Grid Sparse Estimation

Gabriel Peyré
November 08, 2019

Off-the-Grid Sparse Estimation

Talk @Toulouse "Workshop on Computational aspects of Geometry"

Gabriel Peyré

November 08, 2019
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  1. Gabriel Peyré www.numerical-tours.com Off-the-Grid Sparse Estimation Joint work with
 Vincent

    Duval, Quentin Denoyelle Clarice Poon, Nicolas Keriven É C O L E N O R M A L E S U P É R I E U R E
  2. Sparse Estimation Neural spikes (1D) Seismic imaging (1.5D) Astrophysics (2D)

    varying the azimuth φ during the modeled by the following expressio gðθÞ = Z2π 0 Z∞ 0 Z∞ −∞ Iðz; α; φÞρ  Ω where fðzÞ is the density of fluoro volved by the emission point spread the laser beam profile of divergenc describes the intensity of the electr depth z, the incidence angle α, and A B C D Fig. 2. Experimental validation of the m of the system designed to create a slope of the maximum intensity projection of im TIRF illumination. (Scale bar: 5 μm.) The ev illumination angle θ of two selected bead sponding fitting theoretical model (cont depth (respectively 10 and 89 nm). (D) Dep fitting the theoretical TIRF model (in red) estimated by fitting a Gaussian model in t [Boulanger et al. 2014] Single-molecule fluorescence (3-D) Recover pointwise sources from noisy low-resolution observations.
  3. Sparse Estimation Neural spikes (1D) Seismic imaging (1.5D) Astrophysics (2D)

    varying the azimuth φ during the modeled by the following expressio gðθÞ = Z2π 0 Z∞ 0 Z∞ −∞ Iðz; α; φÞρ  Ω where fðzÞ is the density of fluoro volved by the emission point spread the laser beam profile of divergenc describes the intensity of the electr depth z, the incidence angle α, and A B C D Fig. 2. Experimental validation of the m of the system designed to create a slope of the maximum intensity projection of im TIRF illumination. (Scale bar: 5 μm.) The ev illumination angle θ of two selected bead sponding fitting theoretical model (cont depth (respectively 10 and 89 nm). (D) Dep fitting the theoretical TIRF model (in red) estimated by fitting a Gaussian model in t [Boulanger et al. 2014] Single-molecule fluorescence (3-D) Recover pointwise sources from noisy low-resolution observations. Mixture estimation <latexit 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<latexit 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  4. Sparse Estimation Neural spikes (1D) Seismic imaging (1.5D) Astrophysics (2D)

    varying the azimuth φ during the modeled by the following expressio gðθÞ = Z2π 0 Z∞ 0 Z∞ −∞ Iðz; α; φÞρ  Ω where fðzÞ is the density of fluoro volved by the emission point spread the laser beam profile of divergenc describes the intensity of the electr depth z, the incidence angle α, and A B C D Fig. 2. Experimental validation of the m of the system designed to create a slope of the maximum intensity projection of im TIRF illumination. (Scale bar: 5 μm.) The ev illumination angle θ of two selected bead sponding fitting theoretical model (cont depth (respectively 10 and 89 nm). (D) Dep fitting the theoretical TIRF model (in red) estimated by fitting a Gaussian model in t [Boulanger et al. 2014] Single-molecule fluorescence (3-D) Recover pointwise sources from noisy low-resolution observations. Perceptron with 1 hidden layer <latexit 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Mixture estimation <latexit 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<latexit 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  5. Sparse Estimation Neural spikes (1D) Seismic imaging (1.5D) Astrophysics (2D)

    varying the azimuth φ during the modeled by the following expressio gðθÞ = Z2π 0 Z∞ 0 Z∞ −∞ Iðz; α; φÞρ  Ω where fðzÞ is the density of fluoro volved by the emission point spread the laser beam profile of divergenc describes the intensity of the electr depth z, the incidence angle α, and A B C D Fig. 2. Experimental validation of the m of the system designed to create a slope of the maximum intensity projection of im TIRF illumination. (Scale bar: 5 μm.) The ev illumination angle θ of two selected bead sponding fitting theoretical model (cont depth (respectively 10 and 89 nm). (D) Dep fitting the theoretical TIRF model (in red) estimated by fitting a Gaussian model in t [Boulanger et al. 2014] Single-molecule fluorescence (3-D) Recover pointwise sources from noisy low-resolution observations. Practice: Scalable algorithms? Theory: Rayleigh limit? Perceptron with 1 hidden layer <latexit 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Mixture estimation <latexit 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<latexit 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  6. Overview • Sparse Linear Models • Sparse Estimation with Blasso

    • Non-super-resolution Regime • Super-resolution Regime
  7. Sparse Linear Models Measure m on X (e.g. ⇢ Rd

    or R/Z)d). <latexit 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<latexit 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<latexit 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<latexit 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ma,x m Discrete measure: ma,x = Pn i=1 ai xi , <latexit 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<latexit 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ai 2 R, xi 2 X. <latexit 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X = R <latexit 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˜ ' <latexit 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<latexit 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  8. Sparse Linear Models Measure m on X (e.g. ⇢ Rd

    or R/Z)d). <latexit 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<latexit 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<latexit 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<latexit 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ma,x m Discrete measure: ma,x = Pn i=1 ai xi , <latexit 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<latexit 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ai 2 R, xi 2 X. <latexit 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X = R <latexit 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<latexit sha1_base64="qNcoLfo8aoKC1BXk+S6FF9Vg7J4=">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</latexit> Linear model: <latexit 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<latexit 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<latexit sha1_base64="/6//g/fwmJ5K44RZnOFiOnUapso=">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</latexit> y = m def. = R X '(x)dm(x) <latexit 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<latexit 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'(x) 2 H ' continuous. ˜ ' <latexit 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<latexit 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<latexit 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  9. Sparse Linear Models Measure m on X (e.g. ⇢ Rd

    or R/Z)d). <latexit 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<latexit 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<latexit 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<latexit 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ma,x m Discrete measure: ma,x = Pn i=1 ai xi , <latexit 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<latexit 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ai 2 R, xi 2 X. <latexit 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X = R <latexit 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<latexit sha1_base64="qNcoLfo8aoKC1BXk+S6FF9Vg7J4=">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</latexit> Linear model: <latexit 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<latexit 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<latexit sha1_base64="/6//g/fwmJ5K44RZnOFiOnUapso=">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</latexit> y = m def. = R X '(x)dm(x) <latexit 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<latexit 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'(x) 2 H ' continuous. '(x) = ˜ '(· x) Deconvolution: ! Signal-dependent recovery criteria. ma,x <latexit 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<latexit 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˜ ' <latexit 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<latexit 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  10. Sparse Linear Models Measure m on X (e.g. ⇢ Rd

    or R/Z)d). <latexit 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<latexit 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<latexit 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<latexit 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ma,x m Discrete measure: ma,x = Pn i=1 ai xi , <latexit 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<latexit 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ai 2 R, xi 2 X. <latexit 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X = R <latexit 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<latexit sha1_base64="qNcoLfo8aoKC1BXk+S6FF9Vg7J4=">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</latexit> Linear model: <latexit 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<latexit 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<latexit sha1_base64="/6//g/fwmJ5K44RZnOFiOnUapso=">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</latexit> y = m def. = R X '(x)dm(x) <latexit 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<latexit 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<latexit 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'(x) 2 H ' continuous. '(x) = ˜ '(· x) Deconvolution: ! Signal-dependent recovery criteria. Fourier: '(x) = (ei`x)fc `= fc 2 C2fc+1 Laplace: '(x) = e x· 2 H def. = L2(R+) ma,x <latexit 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˜ ' <latexit 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<latexit 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  11. Mixture Models fa,(µ, ) (t) = n X i=1 ai

    d i h ✓ t µi i ◆ <latexit 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<latexit 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Mixture estimation: <latexit 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<latexit sha1_base64="6ojsHTechOlCSQponuc2bRab+Yk=">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</latexit> non-convex <latexit 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<latexit 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fa,(µ, ) <latexit 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<latexit 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<latexit 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Position/scale mixture in Rd: <latexit 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(µ, ) = (mean, std) 2 X def. = Rd ⇥ R+: <latexit 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<latexit 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<latexit 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  12. Mixture Models fa,(µ, ) (t) = n X i=1 ai

    d i h ✓ t µi i ◆ <latexit 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<latexit 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Mixture estimation: <latexit 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<latexit sha1_base64="6ojsHTechOlCSQponuc2bRab+Yk=">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</latexit> non-convex <latexit 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<latexit 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fa,(µ, ) <latexit 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<latexit 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<latexit 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Position/scale mixture in Rd: <latexit 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(µ, ) = (mean, std) 2 X def. = Rd ⇥ R+: <latexit 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<latexit 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<latexit 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  13. Mixture Models fa,(µ, ) (t) = n X i=1 ai

    d i h ✓ t µi i ◆ <latexit 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<latexit 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Mixture estimation: <latexit 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<latexit sha1_base64="6ojsHTechOlCSQponuc2bRab+Yk=">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</latexit> non-convex <latexit 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<latexit 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ma,(µ, ) ! m <latexit 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<latexit sha1_base64="9vMGMwQS3pbbmVhNglwTbIvH5w4=">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</latexit> convex <latexit 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<latexit 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<latexit 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sha1_base64="wswFr9N48PN5PTq6FakwRJoZ8Y0=">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</latexit> where '(x) = 1 xd 2 h ✓ · x1 x2 ◆ <latexit 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<latexit 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<latexit sha1_base64="dHOiuJo4i/VE4OyoMWgNN9q2Xt4=">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</latexit> fa,(µ, ) = ma,(µ, ) def. = Z Rd '(x)dma,(µ, ) (x) <latexit 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<latexit 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Position/scale mixture in Rd: <latexit 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(µ, ) = (mean, std) 2 X def. = Rd ⇥ R+: <latexit 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<latexit 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<latexit 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  14. Multilayer Perceptron fa,(m,s) (t) = n X i=1 ai⇢(hµi, ti

    + i) <latexit 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<latexit 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<latexit 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ai <latexit 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<latexit 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<latexit 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<latexit 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⇢ <latexit 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t 2 Rd <latexit 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<latexit 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<latexit 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<latexit 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(µi, i) <latexit 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fa,(µ, ) (t) 2 R <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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  15. Multilayer Perceptron fa,(m,s) (t) = n X i=1 ai⇢(hµi, ti

    + i) <latexit 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<latexit 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<latexit 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ai <latexit 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<latexit 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<latexit 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<latexit 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⇢ <latexit 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t 2 Rd <latexit 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<latexit 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<latexit 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<latexit 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(µi, i) <latexit 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<latexit 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<latexit 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<latexit sha1_base64="W6/DqUFkF2xG9dsj/T7UnJ+Limg=">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</latexit> min a,(µ, ) R |f(t) fa,(µ, ) (t)|2dt <latexit 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<latexit 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fa,(µ, ) (t) 2 R <latexit 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<latexit 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<latexit 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<latexit sha1_base64="DhRGoOVHGxLQpKZM8C7zNHBYIBg=">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</latexit>
  16. Multilayer Perceptron convex <latexit 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<latexit 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<latexit 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ma,w ! m <latexit 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<latexit 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non-convex <latexit 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<latexit 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<latexit sha1_base64="+0kG9OzlxgMtVWIs0cFmsrS1yJY=">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</latexit> fa,(m,s) (t) = n X i=1 ai⇢(hµi, ti + i) <latexit 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<latexit 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ai <latexit 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<latexit 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<latexit 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<latexit 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⇢ <latexit 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t 2 Rd <latexit 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<latexit 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<latexit 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<latexit 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(µi, i) <latexit 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<latexit 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<latexit 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<latexit sha1_base64="W6/DqUFkF2xG9dsj/T7UnJ+Limg=">AABBw3ictVzbchu5EYU3t7Vz824e8zKJ7JQ35TiSdquSqq1UrSzJsta0TZuU7N2l7eJlRNMacmgOKV+4+oS8Jp+T78gfJE/5hfQFGGBIzDRGcYyShAFxuhs9QKO7Abo3TUbZfHPzn5c++sEPf/Tjn3x8+cpPf/bzX/zy6iefHmfpYtaPj/ppks6e9rpZnIwm8dF8NE/ip9NZ3B33kvhJ73QXP39yFs+yUTppz99N42fj7nAyOhn1u3Noal07ufbi6sbmrU36F61XtnRlQ+l/zfST6Eh11EClqq8WaqxiNVFzqCeqqzIo36kttamm0PZMLaFtBrURfR6rc3UFsAvoFUOPLrSewu8hPH2nWyfwjDQzQveBSwI/M0BG6jpgUug3gzpyi+jzBVHG1jLaS6KJsr2Dvz1Nawytc/USWiWc6RmKw7HM1Yn6M41hBGOaUguOrq+pLEgrKHnkjGoOFKbQhvUBfD6Dep+QRs8RYTIaO+q2S5//i3piKz73dd+F+jdJeR1KpFp69GlOoavOiH5Eb3MBn7E8CXAeAoVYjxFrb0jXYxr9BPovof0BlHOqGZ30oCyp9bwSuQvFh9wVkQdQfMgDEdmA4kM2RGQTig/Z1EjEzkjnfnwLig/fEjk/guJDPhKRj6H4kI9F5DEUH/JYRH4LxYf8VkTegeJD3hGR96D4kPdEZBuKD9kWkUdQfMgjEbkPxYfc18jylTqDkhKdkbAqd6Be5IGWIoGWHVG+22QdfdjbAWu6X4KVV/Ue/PVj9wJ0Gpdg9wPm3UkJVp55B2Aj/VjZFt2l3cSHvStiD2EG+LGHIvZr9aoE+3XASjstwcprrQH9/FjZ+t6HJz/2voh9ADU/Vt6jHkKLH/swYMeYlmCbIvaRel2CDbH6sxKsbPdbYFf8WHmfakN/PzbEmi5KsLI9PQYPxo+Vd6sn0OrHPhGxT9XbEuxTEfsNWHc/9puAHfZ9CdbssVdoBxmSPxLDiq2i1s1XJdamQK0r8E/yvSUh37gH7RJmmGOGhBmLiIMccRCIaOSIRrBcWW5HM/J3ZS6tHNEKRPTyvQlrc7H/IO+PtSQAsZcj9lYQVR4pvmszljPyLkyLhJznOxfWQsaU5vYba7GeD9WW1yAeFhA8t1/SzL9J0RJGUKipKmov8z2ekRE9VyHeUPRmRml4yLh5bhVc1FsR1fOgeiLqnQf1TkQtPKiFiDrzoM5ElF35Lq4TMAOs/vFdLOmJZwD7yOUlAq9gB3adu7BGI5g/TfACH1PLQ/jbothbKlWSYTSP+yRmOZ4VLPEMaku1Ae02Ktyj+DqhFRaDZNzzoY7x8QlzG0u95tgKn+c7eZRnTMLpjEieYU4HvcWI1lM9Oveo5Zy8O67Vw9/N172p1cPvk8bPyYvnWj38XEs/v4DsbY1tXwDbgtU01dq39bo0OP/CNEz9Cu26aHHxrY71nEF6b2vSP9Rv5vAC72WXaqwfW69HI3PGlxXGV4eG1XPm6LkeFfSe2Os1taj2SCY67rX1ujKktItOtBz2qe6bwT4D/WZMvR6NJnhcuxRzL5163dk7zUdj6/VoHCvOe56TJ2/q9WgM6Zn1Yev1aGC2pavjfFuva9lRAxw723pdqz6hLDDmgHjOc4v1imbkJy00tRH5B9XZGtfnX9/HMGfzPI8RqilZ37acTi/fy6olMv5CDFZtXlMO9C8Wjg9WpLFU22J8xTLMC/v7Oh27x6PmG6DFCFY/nwFIOfMEJDQ5CbTeCVDcEqOu4sgMblvE4Sw5WUF1dOtc9BYtX84aFdteUKsUl9nRWj12yF5nNPem5BM2SLOSHhqlb7iMoqShRkFDMr06unuv12tR+5sibrqCmOYzrU8nQnySVh2n+rTecnR8XZ/yzKHwmY+dv5htPtHWBmOelGwRylLF0+1n8khuG+6rN5XNcfNnEb1RtFdnZDVGdCKViVGoyRazN76kZ0v7iM7kkAfT6MN7jDSVqeJTM8yiYz49Iovq2luJN+rLZOi4npHVNfa4Gj100EMPun6Msws7xgOotSFmOIKndkCUcyXXVUoan6k/5KejKb3B6og+KVhIQ4PtTVywkFVR9ssClTeAxtnAUXo4jVU6Bt9ZoyRH/T55bOxatPzX6eTWnG93aY6Xz+byTMyAuG4T14hWDZ/q8tMqB5Zg6f1km/zX6lEivzoc0YZKXJ87nFkvEzrxjymCnZJnnNBqk1ZHsbebn1r9xHBqKnN2jqfZKVnIiOxfBPtTSnMyoh/37oA5QWeLkJCNDLE7o9y78fk6I3GOWT9upPhWg51vMdmyBfE3dN3VldFc5IiB94HzlbltdNIgXzAmrjNt3e3art59EGnvSbizhCnauXKD+H9Gv82PmScbazMCNYxvINO2zvc+UopZUEdd2uWrbZDp60p5LZfhuZba7n9WpmsFyfYo4kJ5cLceAOc+PTMvnCUzkjtb68P7aFU2FylPV/SIoz2hKJ7t/lDvwCj3TdolN2jNdWiWDGEWzPMowvSVssirfKt5FamH0c7+L9StrotaQ4qRshlc1pCU348pWnOlTGBW8/w9pdXk1/pspVc1nwnNxbGzlr+H1t/AbyO3eQ6j0ytYhds0B5iCfbIa4ZZorUcYr9sFXmZmGlr22fKzc9L0clsuEl+zdbMx9lltKk2aNW911sLUL0LjlUPjVaAO23TWaLVo2o0leiHGFm19WhnKrw63dg3KC5Gy7JEZ1ChASjeWCqM6EKnKMb5BvRdpbYq0urBa3dMAd82HIP1rfXV1f5/v7pG6Q75Nnzwwjl8GtEpH5HOZ1upIjSkg5y+0fXVXf4dakHuPLChS5nucuGL41KlP5TyX9Hd6Z0vJzluLYO4tvdF9jI3tUP3zNeSY1kRG69IgvqAesZbflSNasUi3HJ8josx/l3wq9juqY2a3t30nUcGfsPEmryrLiyOFCelfyrwdrkWvh078GlFMuNDedQ9o1X/DSIExJpPg9ywzekO4y/FJAnu0PbKf63aKT/EmjkS3SOql+kuAjeGo1851d26ZEZux/R56otbtW/f1kPklwRwlfhc50evSrjbWPupy5flitLp6lys+V+lhscLX6mNBfdzIwkZ5RUxHfRnMhSWqx4UxIVzqjaKO/PUkryMzn06FUja9DeVipoFtzEuKl6R7oIjweXc3vN7cZ8I4emv0eoR1qXGLRAmzcanOD7iWFrNSl9f2IW69XLkbJc5OVLZTGOrubmHtN1vImKxfoqScDfd2Ze8UohQ5C8MU+opv9JbFhy7NL6Hg70j5okPDMSR32AL/dkftqv0PcBvita5zRjOiFrQFg5XYu6vHWexRraPXDnWXfgiHcB4j0LUk/Yh20rqyM2VZcpd6OP03ZAVmKhaltz3rj8HlIo9knVOd8YzIssmjGSnzXZy6YzEcQkZS5BLOh881pFGcKPOdpnpjMNTlERQ51OFh7jGEvXPbuz4vl1O1vta5hPLgXcCcuBgcnvyVxyq2X4iFmjlv5MNzQOtwUkHd7Bb/6zgMH8upPq9Qbhl91+xVwFvnfrHOyKI/XH/NWG4hs7mcYzjPNB+d9Zb8/Njvi2q9qdQZzYenj/6onQOG11JxHlSWjvHuLLLyhlLBcwGfDKn6j/rHJfnbCK9zGmVy1KFkzinKqZkeMjXzjUvf6MxnITJZOmUyFanZOKJFN2J31aG6Az+7uQdY93Yof5eS/yLW//3ZAbSekPUwWXTOHHSoLabshz1FG9CzvT9bJjHe5eW7vW1owbPwBrXiPd8H1B/v+rYLYyv/Bgmv9fsqVYNCRLJ6umfXVQ9GUDx54xyQ+Z5vRHfpOYvFN8/GAWeL5v7UqkRL+kS+WdArxfccKfs0V6f6rB5PDvCGfTfPD0Xqj9TW1XYe91yJc7OUc3OFc0baKXJ463xWfTerjMuuw2WQ587OdL+U4mx7nledG90r5cJ30Kvxwwr80JGyRdo/pUh4pqqzeYsKmgstk3vCOlEmE8l6wDizm7/v6sj2rILXWcD475Wi7zmSHoAsPcp/R3TCNiN6idbNPknPNx2rM6l3K6TV36N8cXVja/X/MlivHG/f2tq8tfVoe+Or2/r/OfhY/Vr9Vt2ANf4n9RVQa6oj4DBUf1V/U3/f2d853ZntzLnrR5c05leq8G/n/L9mcqDk</latexit> min a,(µ, ) R |f(t) fa,(µ, ) (t)|2dt <latexit 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<latexit 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<latexit sha1_base64="6ojsHTechOlCSQponuc2bRab+Yk=">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</latexit> '(x) = ⇢(hx1, ·i + x2) <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="cwRrbsK94QyZwFRoL0TL2niWBBc=">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</latexit> fa,(µ, ) = ma,(µ, ) def. = Z Rd '(x)dma,(µ, ) (x) <latexit 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<latexit 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where <latexit 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<latexit 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fa,(µ, ) (t) 2 R <latexit 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<latexit 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<latexit 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<latexit 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  17. Overview • Sparse Linear Models • Sparse Estimation with Blasso

    • Non-super-resolution Regime • Super-resolution Regime
  18. Off-the-grid Sparse Regularization Grid-free regularization: total variation of measures: ma,x

    dm(x) = f(x)dx |m|(X) def. = sup R X ⌘dm ; f 2 C(X), ||f||1 6 1 <latexit 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<latexit sha1_base64="t7kJpUDFmCOVwAPh5XESJJBFLoo=">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</latexit> |m|(X) = R |f| = ||f||L1 <latexit 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<latexit 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<latexit 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|ma,x |(X) = ||a||`1 <latexit 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  19. Off-the-grid Sparse Regularization Grid-free regularization: total variation of measures: ma,x

    dm(x) = f(x)dx |m|(X) def. = sup R X ⌘dm ; f 2 C(X), ||f||1 6 1 <latexit 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<latexit sha1_base64="t7kJpUDFmCOVwAPh5XESJJBFLoo=">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</latexit> |m|(X) = R |f| = ||f||L1 <latexit 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<latexit 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<latexit 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|ma,x |(X) = ||a||`1 <latexit 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<latexit 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(P (y)) (P0(y)) ! 0+ Beurling Lasso: <latexit 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<latexit 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<latexit 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[De Castro, Gamboa, 2012] <latexit 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<latexit 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<latexit sha1_base64="eCa7hPIc4xy86Dwfp69cCcmYvpw=">AABB+HictVzNchu5EYY3P7t2/rzJMZdJtE7ZidcrKa4kVVupWlmSZa1lmzYp2btL2zUkR9TYQw49Q8qyab5JDrmlcs0LpCrX5AXyBskpr5D+AQYYEjPAKI5RkjAgvu5GD9DoboDuTZI4n66v//PCB9/69ne+++FHFy997/s/+OGPLn/846M8nWX96LCfJmn2pBfmURKPo8NpPE2iJ5MsCke9JHrce7mNnz8+jbI8Tsed6ZtJ9HQUDsfxcdwPp9D0/PJvurNRPJ6PFkH3OAv7843FfHPRHafZaN5tncTBKPg0eLN4thn8KugmYfBu9O5q98nZteeX19ZvrNO/YLWyIStrQv5rpR8Hh6IrBiIVfTETIxGJsZhCPRGhyKF8IzbEuphA21Mxh7YMajF9HomFuATYGfSKoEcIrS/h9xCevpGtY3hGmjmh+8AlgZ8MkIG4ApgU+mVQR24BfT4jythaRXtONFG2N/C3J2mNoHUqTqDVhVM9fXE4lqk4Fr+jMcQwpgm14Oj6ksqMtIKSB8aopkBhAm1YH8DnGdT7hFR6DgiT09hRtyF9/i/qia343Jd9Z+LfJOUVKIFoy9GnBYVQnBL9gN7mDD5jeRLgPAQKkRwj1l6Trkc0+jH0n0P7fSgLqimd9KDMqXVRi9yGYkNuO5F7UGzIPSfyAIoNeeBEtqDYkC2JRGxGOrfj21Bs+LaT80MoNuRDJ/IRFBvykRN5BMWGPHIiv4ZiQ37tRN6GYkPediLvQrEh7zqRHSg2ZMeJPIRiQx46kbtQbMhdiaxeqRmUlOjEjlW5BfUyD7QUCbRsOeW7RdbRhr3lsab7FVj3qt6Bv3bsjodOowrsrse8O67AumfeHthIO9Zti+7QbmLD3nFi92EG2LH7TuyX4kUF9kuPlfayAuteawfQz451W9978GTH3nNi70PNjnXvUQ+gxY594LFjTCqwLSf2oXhVgfWx+lkF1m3322BX7Fj3PtWB/nasjzWdVWDd9vQIPBg71r1bPYZWO/axE/tEnFVgnzixX4F1t2O/8thh31Zg1R57iXaQIfkjEazYOmphsSqxNgFqoYN/UuwtCfnGPWh3YYYFZkiYkROxVyD2PBEHBeLAW668sKM5+btuLu0C0fZE9Iq9CWtTZ/9B0R9riQdip0DsLCHqPFJ812osp+RdqBYXclrsXFjzGVNa2G+sRXI+1FtehXhQQvDcPqGZf52iJYygUFN11E6KPZ6RAT3XIV5T9KZGqXi4cdPCKpioMyeqZ0H1nKg3FtQbJ2pmQc2cqFML6tSJ0ivfxHU9ZoDWP76LOT3xDGAfuboE4BVswa5zB9ZoAPOnBV7gI2p5AH/bFHu7Sp1kGM3jPolZjqclS5xBbS7WoF1HhTsUXye0wiKQjHs+kDE+PmFuYy7XHFvhRbGTB0XGxJ9OTPIMCzroLQa0nprRuUstC/LuuNYMf6dY96rWDL9LGl+QF8+1ZviplH56Dtk7Ets5B7YNq2kita/rTWlw/oVpqPol2nXR4uJbHck5g/TOGtLfl29m/xzvZZtqrB9db0YjN8aXl8bXhIbWc27ouRkV9J7Y61W1oPFIxjLu1fWmMqS0i46lHPqp6ZvBPgP5ZlS9GY0WeFzbFHPPjXrT2TspRqPrzWgcCc57LsiTV/VmNIb0zPrQ9WY0MNsSyjhf15tadtQAx8663tSqjykLjDkgnvPcor2ijPykmaQWk39Qn60xff7VfQxzNs+KGKGekvZtq+n0ir2sXiLlL0Rg1aYN5UD/Ymb4YGUac7HpjK9Yhmlpf1+lo/d41PwBaDGA1c9nAK6ceQISqpwEWu8EKG44o67yyBRu04nDWXK8hOrK1qnTW9R8OWtUbntOra64TI9W67FL9jqnuTchn/CANOvSw0HlG66i6NLQQUlDbnpNdPdWrtey9teduMkSYlLMtD6dCPFJWn2catN629DxFXnKM4XCZz56/mK2+VhaG4x5UrJFKEsdT7OfyiOZbbivXhc6x82fBfRG0V6dktWI6UQqd0ahKlvM3vicnjXtQzqTQx5Mow/vMZBUJoJPzTCLjvn0gCyqaW9dvFFfKkPH9ZysrrLH9eihgR5a0M1jnG3YMe5DrQMxwyE8dTyinEuFrlLSeCY+LU5HU3qD9RF9UrKQigbbm6hkIeui7JMSldeAxtnAUbo/jWU6Ct9doeSO+m3y6Ni1bPmv0MmtOt8OaY5Xz+bqTMyAuG4S14BWDZ/q8tMyB5Zgbv1kk/zX+lEivyYc0Ya6uD4zOLNexnTiH1EEOyHPOKHV5lod5d5mfmr5E8WpJdTZOZ5mp2QhA7J/AexPKc3JgH7MuwPqBJ0tQkI20sfuxIV3Y/N1Yucc035cLPhWg55vEdmyGfFXdM3VldNc5IiB94HF0txWOjkgXzAirpm07npt1+8+iNT3JMxZwhT1XLlK/K/Rb/Wj5snayoxADeMbyKWts72PlGIW1FFIu3y9DVJ9TSk/KWR4JqXW+5+W6ZOSZDsUcaE8uFsPgHOfnpkXzpKM5M5X+vA+WpfNRcqTJT3iaI8pime7P5Q7MMp9nXbJNVpzXZolQ5gF0yKKUH1dWeRlvvW8ytT9aOf/F+pa12WtIcVA6Awua8iV348oWjOlTGBW8/x9SavJrvVsqVc9nzHNxZGxlt9B68/gt5JbPfvR6ZWswi2aA0xBP2mNcEuw0sOP160SLzUzFS39rPnpOal6mS3nia/ZuukY+7QxlRbNmjOZtVD189B4YdB44anDDp01ai2qdmWJnjtji448rfTl14RbpwHlmZOy2yNTqNhDSjOW8qM6cFJ1x/gK9dZJa91JK4TVap4GmGveB2lf68ur+12xuwfiNvk2ffLAOH4Z0CqNyedSrfWRGlNAzjelfTVXf5dakHuPLChS5nucuGL41KlPZVFI+gu5s6Vk57VFUPeWXss+ysZ2qf7rFeSI1kRO61IhblKPSMpvyhEsWaQbhs8RUOY/JJ+K/Y76mNnsrd9JUPIndLzJq0rz4khhTPp3Zd72V6LXfSN+DSgmnEnvuge0mr9hpMAYlUmwe5Y5vSHc5fgkgT3aHtnPVTvFp3hjQ6IbJPVc/N7DxnDUq+e6ObfUiNXYfgk9Uev6rdt6uPkl3hxd/M5zohfSrjaSPup86fl8tEK5y5Wf6/QwW+Kr9TGjPmZkoaO8MqYrPvfmwhI148IYHy7NRtFE/maSN5GZT6d8KaveinI508A25oTiJdc9UETYvLurVm/ummMcvRV6PcKa1LjFRQmzcanMD5iWFrNSF1f2IW69WLsbJcZOVLVTKOrmbqHtN1vIiKxfIlw5G+5tyt4tRSnuLAxT6Au+0VsVH5o0P4eCvwNhiw4VR5/cYRv82y2xLXbfw22IV7LOGc2AWtAWDJZi71COs9yjXkevDOomfR8O/jxi0LVL+ph20qayM2W35CZ1f/qvyQpkInJKr3s2H4PJxT2SVU5NxhOTZXOPJhbquzhNx6I4+IykzMWfD59ruEZxLNR3mpqNQVF3j6DMoQkPdY/B753r3s15mZzq9bXKxZcH7wLqxEXh8OSvOlbR/XwsVGa8kffPAa3DcQ11tVv8r+NQfDSn5rx8ueX0XbMXHm+d+0UyI4v+cPM1o7n5zOZqjv4802J02luy82O/L2j0plJjNO+fPvqjeg4oXnPBeVC3dIw3Z5GW15cKngvYZEjFf8RfL7i/jfCqoFElRxNK6pyimprq4aamvnFpG536zEcmTadKpjI1HUe06UbsttgXt+Fnu/AAm94O5e9S8l/E2r8/O4DWY7IeKovOmYMutUWU/dCnaAN61vdnqyTGu7x8t7cDLXgWfkCteM/3PvXHu76d0tiqv0HCa/2eSMWgFJEsn+7pddWDEZRP3jgHpL7nG9Bdes5i8c2zkcfZoro/tSzRnD5x3yzoVeJ7hpR9mqsTeVaPJwd4wz4s8kOB+IzaQmnncc91cW5Vcm4tcc5JO2UOZ8Zn9XezqrhsG1wGRe7sVPZLKc7W53n1udGdSi58B70eP6zBDw0p26T9lxQJZ6I+mzeroTmTMpknrGOhMpGsB4wzw+J910e2pzW8Tj3Gf7cSfdeQdA9k6VH+O6ATtozoJVI3uyQ933Ssz6TeqZFWfo/y+eW1jeX/y2C1crR5Y2P9xsbDm2tf3JL/z8FH4qfi5+IqrPHfii+AWkscAoc/iL+Jv4t/bL3d+uPWn7b+zF0/uCAxPxGlf1t/+S9jyrVy</latexit> min m {|m|(X) ; m = y} <latexit 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<latexit 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<latexit 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  20. Off-the-grid Sparse Regularization Grid-free regularization: total variation of measures: ma,x

    dm(x) = f(x)dx [Fischer Jerome, 1974] If dim(Im( )) < +1, 9(a, x) 2 RN ⇥ TN with N 6 dim(Im( )) such that ma,x is a solution to P (y). Proposition: |m|(X) def. = sup R X ⌘dm ; f 2 C(X), ||f||1 6 1 <latexit 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<latexit 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<latexit 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<latexit sha1_base64="t7kJpUDFmCOVwAPh5XESJJBFLoo=">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</latexit> |m|(X) = R |f| = ||f||L1 <latexit 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<latexit 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<latexit 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|ma,x |(X) = ||a||`1 <latexit 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<latexit 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(P (y)) (P0(y)) ! 0+ Beurling Lasso: <latexit 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<latexit 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<latexit 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[De Castro, Gamboa, 2012] <latexit 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<latexit 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<latexit sha1_base64="eCa7hPIc4xy86Dwfp69cCcmYvpw=">AABB+HictVzNchu5EYY3P7t2/rzJMZdJtE7ZidcrKa4kVVupWlmSZa1lmzYp2btL2zUkR9TYQw49Q8qyab5JDrmlcs0LpCrX5AXyBskpr5D+AQYYEjPAKI5RkjAgvu5GD9DoboDuTZI4n66v//PCB9/69ne+++FHFy997/s/+OGPLn/846M8nWX96LCfJmn2pBfmURKPo8NpPE2iJ5MsCke9JHrce7mNnz8+jbI8Tsed6ZtJ9HQUDsfxcdwPp9D0/PJvurNRPJ6PFkH3OAv7843FfHPRHafZaN5tncTBKPg0eLN4thn8KugmYfBu9O5q98nZteeX19ZvrNO/YLWyIStrQv5rpR8Hh6IrBiIVfTETIxGJsZhCPRGhyKF8IzbEuphA21Mxh7YMajF9HomFuATYGfSKoEcIrS/h9xCevpGtY3hGmjmh+8AlgZ8MkIG4ApgU+mVQR24BfT4jythaRXtONFG2N/C3J2mNoHUqTqDVhVM9fXE4lqk4Fr+jMcQwpgm14Oj6ksqMtIKSB8aopkBhAm1YH8DnGdT7hFR6DgiT09hRtyF9/i/qia343Jd9Z+LfJOUVKIFoy9GnBYVQnBL9gN7mDD5jeRLgPAQKkRwj1l6Trkc0+jH0n0P7fSgLqimd9KDMqXVRi9yGYkNuO5F7UGzIPSfyAIoNeeBEtqDYkC2JRGxGOrfj21Bs+LaT80MoNuRDJ/IRFBvykRN5BMWGPHIiv4ZiQ37tRN6GYkPediLvQrEh7zqRHSg2ZMeJPIRiQx46kbtQbMhdiaxeqRmUlOjEjlW5BfUyD7QUCbRsOeW7RdbRhr3lsab7FVj3qt6Bv3bsjodOowrsrse8O67AumfeHthIO9Zti+7QbmLD3nFi92EG2LH7TuyX4kUF9kuPlfayAuteawfQz451W9978GTH3nNi70PNjnXvUQ+gxY594LFjTCqwLSf2oXhVgfWx+lkF1m3322BX7Fj3PtWB/nasjzWdVWDd9vQIPBg71r1bPYZWO/axE/tEnFVgnzixX4F1t2O/8thh31Zg1R57iXaQIfkjEazYOmphsSqxNgFqoYN/UuwtCfnGPWh3YYYFZkiYkROxVyD2PBEHBeLAW668sKM5+btuLu0C0fZE9Iq9CWtTZ/9B0R9riQdip0DsLCHqPFJ812osp+RdqBYXclrsXFjzGVNa2G+sRXI+1FtehXhQQvDcPqGZf52iJYygUFN11E6KPZ6RAT3XIV5T9KZGqXi4cdPCKpioMyeqZ0H1nKg3FtQbJ2pmQc2cqFML6tSJ0ivfxHU9ZoDWP76LOT3xDGAfuboE4BVswa5zB9ZoAPOnBV7gI2p5AH/bFHu7Sp1kGM3jPolZjqclS5xBbS7WoF1HhTsUXye0wiKQjHs+kDE+PmFuYy7XHFvhRbGTB0XGxJ9OTPIMCzroLQa0nprRuUstC/LuuNYMf6dY96rWDL9LGl+QF8+1ZviplH56Dtk7Ets5B7YNq2kita/rTWlw/oVpqPol2nXR4uJbHck5g/TOGtLfl29m/xzvZZtqrB9db0YjN8aXl8bXhIbWc27ouRkV9J7Y61W1oPFIxjLu1fWmMqS0i46lHPqp6ZvBPgP5ZlS9GY0WeFzbFHPPjXrT2TspRqPrzWgcCc57LsiTV/VmNIb0zPrQ9WY0MNsSyjhf15tadtQAx8663tSqjykLjDkgnvPcor2ijPykmaQWk39Qn60xff7VfQxzNs+KGKGekvZtq+n0ir2sXiLlL0Rg1aYN5UD/Ymb4YGUac7HpjK9Yhmlpf1+lo/d41PwBaDGA1c9nAK6ceQISqpwEWu8EKG44o67yyBRu04nDWXK8hOrK1qnTW9R8OWtUbntOra64TI9W67FL9jqnuTchn/CANOvSw0HlG66i6NLQQUlDbnpNdPdWrtey9teduMkSYlLMtD6dCPFJWn2catN629DxFXnKM4XCZz56/mK2+VhaG4x5UrJFKEsdT7OfyiOZbbivXhc6x82fBfRG0V6dktWI6UQqd0ahKlvM3vicnjXtQzqTQx5Mow/vMZBUJoJPzTCLjvn0gCyqaW9dvFFfKkPH9ZysrrLH9eihgR5a0M1jnG3YMe5DrQMxwyE8dTyinEuFrlLSeCY+LU5HU3qD9RF9UrKQigbbm6hkIeui7JMSldeAxtnAUbo/jWU6Ct9doeSO+m3y6Ni1bPmv0MmtOt8OaY5Xz+bqTMyAuG4S14BWDZ/q8tMyB5Zgbv1kk/zX+lEivyYc0Ya6uD4zOLNexnTiH1EEOyHPOKHV5lod5d5mfmr5E8WpJdTZOZ5mp2QhA7J/AexPKc3JgH7MuwPqBJ0tQkI20sfuxIV3Y/N1Yucc035cLPhWg55vEdmyGfFXdM3VldNc5IiB94HF0txWOjkgXzAirpm07npt1+8+iNT3JMxZwhT1XLlK/K/Rb/Wj5snayoxADeMbyKWts72PlGIW1FFIu3y9DVJ9TSk/KWR4JqXW+5+W6ZOSZDsUcaE8uFsPgHOfnpkXzpKM5M5X+vA+WpfNRcqTJT3iaI8pime7P5Q7MMp9nXbJNVpzXZolQ5gF0yKKUH1dWeRlvvW8ytT9aOf/F+pa12WtIcVA6Awua8iV348oWjOlTGBW8/x9SavJrvVsqVc9nzHNxZGxlt9B68/gt5JbPfvR6ZWswi2aA0xBP2mNcEuw0sOP160SLzUzFS39rPnpOal6mS3nia/ZuukY+7QxlRbNmjOZtVD189B4YdB44anDDp01ai2qdmWJnjtji448rfTl14RbpwHlmZOy2yNTqNhDSjOW8qM6cFJ1x/gK9dZJa91JK4TVap4GmGveB2lf68ur+12xuwfiNvk2ffLAOH4Z0CqNyedSrfWRGlNAzjelfTVXf5dakHuPLChS5nucuGL41KlPZVFI+gu5s6Vk57VFUPeWXss+ysZ2qf7rFeSI1kRO61IhblKPSMpvyhEsWaQbhs8RUOY/JJ+K/Y76mNnsrd9JUPIndLzJq0rz4khhTPp3Zd72V6LXfSN+DSgmnEnvuge0mr9hpMAYlUmwe5Y5vSHc5fgkgT3aHtnPVTvFp3hjQ6IbJPVc/N7DxnDUq+e6ObfUiNXYfgk9Uev6rdt6uPkl3hxd/M5zohfSrjaSPup86fl8tEK5y5Wf6/QwW+Kr9TGjPmZkoaO8MqYrPvfmwhI148IYHy7NRtFE/maSN5GZT6d8KaveinI508A25oTiJdc9UETYvLurVm/ummMcvRV6PcKa1LjFRQmzcanMD5iWFrNSF1f2IW69WLsbJcZOVLVTKOrmbqHtN1vIiKxfIlw5G+5tyt4tRSnuLAxT6Au+0VsVH5o0P4eCvwNhiw4VR5/cYRv82y2xLXbfw22IV7LOGc2AWtAWDJZi71COs9yjXkevDOomfR8O/jxi0LVL+ph20qayM2W35CZ1f/qvyQpkInJKr3s2H4PJxT2SVU5NxhOTZXOPJhbquzhNx6I4+IykzMWfD59ruEZxLNR3mpqNQVF3j6DMoQkPdY/B753r3s15mZzq9bXKxZcH7wLqxEXh8OSvOlbR/XwsVGa8kffPAa3DcQ11tVv8r+NQfDSn5rx8ueX0XbMXHm+d+0UyI4v+cPM1o7n5zOZqjv4802J02luy82O/L2j0plJjNO+fPvqjeg4oXnPBeVC3dIw3Z5GW15cKngvYZEjFf8RfL7i/jfCqoFElRxNK6pyimprq4aamvnFpG536zEcmTadKpjI1HUe06UbsttgXt+Fnu/AAm94O5e9S8l/E2r8/O4DWY7IeKovOmYMutUWU/dCnaAN61vdnqyTGu7x8t7cDLXgWfkCteM/3PvXHu76d0tiqv0HCa/2eSMWgFJEsn+7pddWDEZRP3jgHpL7nG9Bdes5i8c2zkcfZoro/tSzRnD5x3yzoVeJ7hpR9mqsTeVaPJwd4wz4s8kOB+IzaQmnncc91cW5Vcm4tcc5JO2UOZ8Zn9XezqrhsG1wGRe7sVPZLKc7W53n1udGdSi58B70eP6zBDw0p26T9lxQJZ6I+mzeroTmTMpknrGOhMpGsB4wzw+J910e2pzW8Tj3Gf7cSfdeQdA9k6VH+O6ATtozoJVI3uyQ933Ssz6TeqZFWfo/y+eW1jeX/y2C1crR5Y2P9xsbDm2tf3JL/z8FH4qfi5+IqrPHfii+AWkscAoc/iL+Jv4t/bL3d+uPWn7b+zF0/uCAxPxGlf1t/+S9jyrVy</latexit> min m {|m|(X) ; m = y} <latexit 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<latexit 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<latexit 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  21. Off-the-grid Sparse Regularization Grid-free regularization: total variation of measures: ma,x

    dm(x) = f(x)dx [Fischer Jerome, 1974] If dim(Im( )) < +1, 9(a, x) 2 RN ⇥ TN with N 6 dim(Im( )) such that ma,x is a solution to P (y). Proposition: Questions: is m0 solution of P0( m0)? Robustness to noise? <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="4X5prpF66D9UHrXlrlETqmis1OQ=">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</latexit> |m|(X) def. = sup R X ⌘dm ; f 2 C(X), ||f||1 6 1 <latexit 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<latexit 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<latexit sha1_base64="t7kJpUDFmCOVwAPh5XESJJBFLoo=">AABCIXictVxLcxu5EYY3r7Xy8ibHXJDITtkpR5GUrUqqtlK1MiXLWsu2bFKyd5e2io8hTXvIoTikLJvmr8kPySmH3FK5pfIHksohl+Se7gYwwJCYaYziGCUJA+LrbvQAje4G6PY4HqTTzc2/XvnoG9/81re/8/HVte9+7/s/+OG1T350kiazSSc67iRxMnnWbqVRPBhFx9PBNI6ejSdRa9iOo6ft1zX8/Ol5NEkHyagxfTuOng9b/dGgN+i0ptB0eu30+vvh+5vNZxe3ZDM660Y92UxnY6iP0k4y6s5lczCansLn0DRtyWZXDuVivtbDdtmsdQh6WzZHyWQ47y1Oobk3fSubcXQmt9YW10+vrW9ubNI/uVrZ0pV1of8dJZ/IY9EUXZGIjpiJoYjESEyhHouWSKF8LbbEphhD23Mxh7YJ1Ab0eSQWYg2wM+gVQY8WtL6G3314+lq3juAZaaaE7gCXGH4mgJTiBmAS6DeBOnKT9PmMKGNrEe050UTZ3sLftqY1hNapeAmtHM70DMXhWKaiJ35LYxjAmMbUgqPraCoz0gpKLp1RTYHCGNqw3oXPJ1DvENLoWRImpbGjblv0+d+pJ7bic0f3nYl/kJQ3oEhR16NPMgotcU70Jb3NGXym5ImBcx8oRHqMWHtDuh7S6EfQfw7tD6EsqGZ00oYyp9ZFKbIGxYessch9KD7kPos8hOJDHrLIIyg+5JFGInZCOvfj61B8+DrL+TEUH/Ixi3wCxYd8wiJPoPiQJyzyKyg+5Fcs8i4UH/Iui7wPxYe8zyIbUHzIBos8huJDHrPIPSg+5J5GFq/UCZSE6AyYVbkD9TwPtBQxtOyw8t0h6+jD3glY050CLL+qd+GvH7sboNOoALsXMO96BVh+5u2DjfRjeVt0j3YTH/Yeiz2AGeDHHrDYL8SrAuwXASvtdQGWX2uH0M+P5a3vA3jyYx+w2IdQ82P5PeoRtPixjwJ2jHEB9ojFPhZnBdgQqz8pwPJ2vw52xY/l96kG9PdjQ6zprADL29MT8GD8WH63egqtfuxTFvtMXBRgn7HYL8G6+7FfBuyw7wqwZo9dox2kT/5IBCu2jForW5VYGwO1FsM/zvaWmHzjNrRzmH6G6RNmyCL2M8R+IOIwQxwGy5VmdjQlf5fnUs8Q9UBEO9ubsDZl+3ez/liLAxC7GWJ3CVHmkeK7NmM5J+/CtHDIabZzYS1kTElmv7EW6flQbnkN4lEOoeb2S5r5tylawggKNVVG7WW2xyukpOcyxBuK3swoDQ8eN82sgou6YFFtD6rNot56UG9Z1MyDmrGocw/qnEXZle/imgEzwOof38WcntQMUD5ycZHgFezArnMP1qiE+XMEXuATankEf+sUe3OlTDKM5nGfxCzH85wlnkBtLtah3UaFuxRfx7TCIpBM9XykY3x8wtzGXK85ZYUX2U4us4xJOJ0BydPP6KC3KGk9VaNzn1oW5N2pWjX8vWzdm1o1/B5pfEFevKpVw0+19NNLyN7Q2MYlsHVYTWOtfVuvSkPlXxQNU1+jXRctLr7VoZ4zSO+iIv0D/WYOLvFealRT+rH1ajRSZ3xpbnxVaFg9p46eq1FB70l5vaYmK49kpONeW68qQ0K76EjLYZ+qvhns09VvxtSr0TgCj6tGMffcqVedveNsNLZejcaJUHnPBXnypl6NRp+elT5svRoNzLa0dJxv61UtO2pAxc62XtWqjygLjDkgNedVi/WKJuQnzTS1AfkH5dka1+df3ccwZ/MiixHKKVnftphOO9vLyiUy/kIEVm1aUQ70L2aOD5anMRfbbHylZJjm9vdVOnaPR80fghYlrH51BsDlzGOQ0OQk0HrHQHGLjbryIzO4bRaHs6S3hGrq1inrLVq+KmuUbzulVi4us6O1emySvU5p7o3JJzwkzXJ6OCx8w0UUOQ0d5jTE06uiu3d6vea1v8nixkuIcTbTOnQipE7SyuNUn9brjo5v6FOeKRR15mPnL2abe9raYMyTkC1CWcp4uv1MHsltw331trA5bvWZpDeK9uqcrMaATqRSNgo12WLljc/p2dI+pjM55KFodOA9Sk1lLNSpGWbRMZ8uyaK69pbjjfoyGTpVT8nqGntcju476L4HXT3GqcGO8RBqDYgZjuGpERDlrGW6SkjjE/HL7HQ0oTdYHtHHOQtpaCh7E+UsZFmU/TJH5Q2gcTaoKD2cxjIdg2+uUOKjfp88NnbNW/4bdHJrzrdbNMeLZ3NxJqZLXLeJq6RVo0511dMyByXB3PvJNvmv5aNEflU4og3luL5wOCu9jOjEP6IIdkyecUyrjVsd+d5ufmr5E8PpSJizczzNTshCSrJ/EvanhOakpB/37oA5QVcWISYbGWJ3Bpl34/N1Buwcs37cQKhbDXa+RWTLZsTf0HVXV0pzUUUMah9YLM1to5ND8gUj4jrR1t2u7fLdB5H2noQ7SxRFO1duEv9b9Nv8mHmyvjIjUMP4BlJt63zvI6GYBXXUol2+3AaZvq6U1zMZXmip7f5nZbqek2yXIi6UB3frLnDu0LPihbNkQnKnK33UPlqWzUXK4yU94mh7FMUru9/XOzDKfZt2yXVac02aJX2YBdMsijB9uSzyMt9yXnnqYbTT/wt1q+u81pCiFDaDqzTE5fcjitZcKWOY1Wr+vqbV5Nf6ZKlXOZ8RzcWhs5bfQ+tP4beR2zyH0WnnrMIdmgOKgn2yGlEtcqVHGK87OV5mZhpa9tnys3PS9HJbLhNfK+tmY+zzylSOaNZc6KyFqV+GxiuHxqtAHTborNFq0bQbS3TKxhYNfVoZyq8Kt0YFyjOWMu+RGdQgQEo3lgqj2mWp8jG+Qb1jaW2ytFqwWt3TAHfNhyD9a315db/Pdncp7pJv0yEPTMUvXVqlA/K5TGt5pKYoIOdPtX11V3+TWpB7mywoUlb3OHHFqFOnDpVFJunP9c6WkJ23FsHcW3qj+xgb26T6r1eQQ1oTKa1Lg/iUekRaflcOuWSRNhyfQ1Lmv0U+lfI7ymNmt7d9JzLnT9h4U60qy0tFCiPSP5d5O1iJXg+c+FVSTDjT3nUbaFV/w0hBYUwmwe9ZpvSGcJdTJwnKo22T/Vy1U+oUb+RItEFSz8XvAmyMinrtXHfnlhmxGdsvoCdq3b51Xw+eXxzMkeN3mRO9Fu1qQ+2jzpeeL0erpXe5/HOZHmZLfK0+ZtTHjSxslJfHNMVnwVyURNW4KEwIl2qjqCJ/NcmryKxOp0Ipm96Gcj7ToGzMS4qXuHugiPB5dze93twtZhztFXptwrrUVAtHCbNxic4PuJYWs1JXV/Yh1Xq1dDeKnZ2oaKcw1N3dwtpvZSEjsn6x4HI2qrcrezMXpfBZGEWhI9SN3qL40KX5GRT8LYUvOjQcQ3KHdfBvd0RN7H2A2xBnuq4ympJa0BZ0l2Lvlh5nvke5js4c6i79EA7hPAaga076Ae2kVWVXlHnJXerh9N+QFZiIiJXe9qw+BpcLP5JVTlXGMyDLxo9mIMx3caqOxXAIGUmeSzgfda7BjaInzHeaqo3BUOdHkOdQhYe5xxD2zm3v6rxcTuX6WuUSykPtAubExeDw5K84VrH9QizUxHkjH54DWodeCXWzW/yv4zB8LKfqvEK5pfRds1cBb131i3RGFv3h6mvGcguZzcUcw3km2eist+Tnp/w+WelNJc5oPjx99EftHDC85kLlQXnpFN6dRVbeUCp4LuCTIRH/En+4wn8b4SyjUSRHFUrmnKKYmunBUzPfuPSNznwWIpOlUyRTnpqNI+p0I7YmDsRd+KllHmDV26Hqu5TqL2L935/tQmuPrIfJoqvMQZPaIsp+2FO0Lj3b+7NFEuNdXnW3twEteBZ+SK14z/ch9ce7vo3c2Iq/QaLW+gORiG4uIlk+3bPrqg0jyJ+8qRyQ+Z6vpLv0Koulbp4NA84Wzf2pZYnm9Al/s6BdiG87UnZoro71WT2eHOAN+1aWH5LiV9TW0nYe91yO81Eh56MlzilpJ8/hwvms/G5WEZeaw6Wb5c7Odb+E4mx7nleeG90t5KLuoJfj+yX4viNlnbT/miLhiSjP5s1KaM60TO4J60iYTKTSA8aZrex9l0e25yW8zgPGf78Qfd+RdB9kaVP+W9IJ24ToxVo3eyS9uulYnkm9VyKt/h7l6bX1reX/y2C1crK9sbW5sfV4e/3zO/r/OfhY/ET8TNyENf4b8TlQOxLHwOGP4p/i3+I/O7/f+dPOn3f+orp+dEVjfixy/3b+9l92WMSL</latexit> |m|(X) = R |f| = ||f||L1 <latexit 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<latexit 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<latexit 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|ma,x |(X) = ||a||`1 <latexit 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<latexit 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(P (y)) (P0(y)) ! 0+ Beurling Lasso: <latexit 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<latexit 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<latexit 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[De Castro, Gamboa, 2012] <latexit 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<latexit 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<latexit sha1_base64="eCa7hPIc4xy86Dwfp69cCcmYvpw=">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</latexit> min m {|m|(X) ; m = y} <latexit 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<latexit 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<latexit 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  22. = 1/fc = 0.6/fc Limit Certificate P (y) Proposition: ⌘

    ⌘ min m |m|(X) + 1 2 || m y||2 <latexit 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<latexit 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m solves (P (y)) , ⌘ 2 @|m |(X) <latexit 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</latexit> <latexit 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</latexit> <latexit 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</latexit> <latexit 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vcIsF559v1SyQ0tDVrnwpsK1imHrPLzHOAx/HsCf1365Zo60Xl4FvWCKPr066IUeuL15NaVK82VEo7qHqod9UgdQRvfVm8a1fG89ETJLv1U2TdecS2ZlJl6l/Ah/zUiHcXSXLVo+qj2Ib4v3nV2DPJgrS9YtWTPHZRzX5CtsA+U/NWyVlFtK56bkVMI5TrkOL7+g1LxHBLPu/M4GHdr8kmV4s1wWUE+mQkpy2aXtKj1e9B2WO+udt0mTtJv2Ct6XTWxrH2+BEWkLaO5GWNGKy9ltKFhcpbrqtpaXdoo34xX18Jsu5WVqrq29kOxtP/vdlYl/6JD/u/SwjiyqufL7lRY2G9t53/bdt7EQ915Iwu6ra/6rS39X7Wl2/oh2478uzC6VMN8o5frvElWumWupW+Zk7u3QuPAIgfZ855aMVPDSpUVf7NCVsQXYxHfHtfb8TPRVd07mWPG4HziQO7aNHfi9vUKFMdvbZpnEHueUTw3odbBd/C2lJyCfV/J3cjVyNEc6r1CdFzGtIjbhDRaRr6bjyeL95HivmicQbBPb8jsYaJPYuHtdGipSQF5P6cn9hWiW6bGt3fL7N2opGE+3WFuEE4K5Svmlb22HD0muURGRtk5fZVLl3jKzfuiRCqhXuTPdyM3KSdSv5fTt2UzEnRoxbJKVwlZtU+b5sTLuuK95z2qYz6BMlW4q3bq4FjN6c05NEknPI9l+1GZNyhr2OSf97xMkf3okDxDDEWTf55i1SxSudzYPta1d04qUW5t2thqjlV1emPFzyHOMTREi246odpCSmtRlEK1VKTk8saJrkeUpUutvE1Wyfv2JjRmyLSf5R1DqN3wPeo3lfmOCz6R7/ofkB5xN0RLl7mKRzln9TkPPl8xIZR5N2fwpI1tqVPauZvo2Sic673ylLJPs20T1dXrOKLZffVQPVJ/DbS6kPsrbykmOo9gk7n9WP79FIf5bX72DuHw7Ugb6ll+R2PVPkJ551HKM33Pm5/uToDqPN2dnK6fckhen8R+yg/VUxo31NEGS30asaP4Ed295qOMN82dUq7G3A351XS3QIaHc3UvqaEdSbJXim+CFmns2wCX6AQh37uAdjipJV2L0uK4mJy34XSgZ25d57fMdyFd7mlddvQtQL7cV3pXZGgv5BWVgnO+VLIvdxyknyk5jylj7Cs6my0jahMvhKkUzwyZk5RcBpumjNtDdK9y+Xw2zXn4zrOZKt8n5rL0GJ5yJxLro0i1rOEYarsFemy3vDe7LHE8daZ9kJfc0C5SXM65zahccfV5oEYW3WVArkTak/i3sg3EcOVzRfbnZUtX13RbV5XuwvLJbwYN6fZEY61VtoW3L7paQx2vYXjezMkwy+2sbLVxFB84adr6uvHaWng3abXM0h/5KBxBX8M3Hh9F9F2c0/RTeJdosTWExxpCZYPOXBw7aJwEqTzIqaAFdDWlIp2TiNLz2LR47sh9Jtt/WrWV37WKt5+eUg3crV2bB0p+q8F8wu999PwUeec2j2mlzXesU75rOkLi+MacksLzH/co7QU8bKWn+ix4ov5qThIbPaWRx0t1RyW5T/HXAd8JJT2pnbZsUcLx1UqUfdnW9STSmowt+fNu5nn3VNwNh4gQjH3bQJFCvLT7MKpz02Pp4ykdaTqhlr+f8yuOqEbE7UkA/Zmqumv2M3Xj7A3FYni+SH4ByP51nhEhzGkW5l3Mj78YhCl9sOQBxb9TVe/Xm5AC7+RB1Da9dfWuH/YNQxo9Y4Qmd0KY3x/i0+fY0nDWyn82DU+dD5Vrlrj8WyQ878Nnc1BKPmd2DGUa0ln3D7THOonwDgOiUL4dx+bQojkm3lmAY+eH6m+DVF+Rbpqlk1kh+Z/kqKRUhtAZCPTWfNYgxM8uzTG0Sxs7z1Vuf1oD+9kFv1fv1FRH61ViiPCJ1fOC/MYXm1sgTrW3akXUbNV91jG/13CQt/f5u1+qboxeBkxo1How50dsGjEUpqp8fzS3j2mwVHKLYRkZ3nXYUeU72LleOhHW2aNxXa/UFooSPKCyr+pRLNb03Tyi81F/QfdPs3++CPZbL6z8+HYKvYj7zGvx+5jZA4kGOGovW6kPfa1jx2Ua37hiBT/2UqPrxR/huEp+w6McC7r6Knfs5reLgZLzweWbnH0nnWS1aDvn4J5H7CjzC5tMlc/yNigtpdo2Vtihz/rs/+nbi2vlX3qdf3m+fn9t9f7aJ+uLv3mgfwX2p+rP1F+ALtbU++o36jHY5BHI9I/qX9W36j8v/+HyXy7/7fI/OOuPf6Qxf6oK/13+1/8AJBBX/Q==</latexit> <latexit sha1_base64="csHHCFGrGncxx727R6lo1yAaHis=">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</latexit>
  23. = 1/fc = 0.6/fc Limit Certificate P (y) ⌘0 def.

    = argmin ⌘= ⇤p ||p|| s.t. ⇢ 8 i, ⌘(xi) = sign(ai), ||⌘||1 6 1. Proposition: Theorem: m ! m0 = ma,x If ( , ||w||/ ) ! 0, then ⌘ ! ⌘0. ⌘ ⌘ min m |m|(X) + 1 2 || m y||2 <latexit 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m solves (P (y)) , ⌘ 2 @|m |(X) <latexit 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</latexit> <latexit 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</latexit> <latexit 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</latexit> <latexit 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</latexit> <latexit sha1_base64="csHHCFGrGncxx727R6lo1yAaHis=">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</latexit>
  24. −1 1 η 0 η V ⌘0 6= ⌘V −1

    1 η 0 η V ⌘0 = ⌘V Vanishing Derivative Pre-certificate Input measure: m0 = ma,x. ⌘0 def. = argmin ⌘= ⇤p ||p|| s.t. ⇢ 8 i, ⌘(xi) = sign(ai), ||⌘||1 6 1. ⌘V def. = argmin ⌘= ⇤p ||p|| s.t. ⇢ 8 i, ⌘(xi) = sign(ai), 8 i, ⌘0(xi) = 0. where Ax(b) = P i b1 i '(xi, ·) + b2 i '0(xi, ·) Proposition: ⌘V = ⇤A+ x (sign(a); 0)
  25. −1 1 η 0 η V ⌘0 6= ⌘V −1

    1 η 0 η V ⌘0 = ⌘V Vanishing Derivative Pre-certificate Theorem: ⌘V 2 ND(m0) =) ⌘V = ⌘0 Input measure: m0 = ma,x. ⌘0 def. = argmin ⌘= ⇤p ||p|| s.t. ⇢ 8 i, ⌘(xi) = sign(ai), ||⌘||1 6 1. ⌘V def. = argmin ⌘= ⇤p ||p|| s.t. ⇢ 8 i, ⌘(xi) = sign(ai), 8 i, ⌘0(xi) = 0. () Non-degenerate certificate: ⌘ 2 ND(ma,x) : 8 t / 2 {x1, . . . , xN }, |⌘(t)| < 1 and 8 i, ⌘00(xi) 6= 0 where Ax(b) = P i b1 i '(xi, ·) + b2 i '0(xi, ·) Proposition: ⌘V = ⇤A+ x (sign(a); 0)
  26. Support Stability Theorem Theorem: the solution of P (y) for

    y = (m0) + w is for (||w||/ , ) = O(1), [Duval, Peyr´ e 2014] If ⌘V 2 ND(m0) for m0 = ma,x, then m = PN i=1 a? i x? i where ||(x, a) (x? , a?)|| = O(||w||) ||w||
  27. When is Non-degenerate ? ⌘V ⌘V ⌘V ⌘V ⌘V ⌘V

    ⌘V Input measure: m0 = ma, x, ! 0 '(0) '(0) Intuition: for signed measures, spikes should be “separated”. <latexit 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<latexit 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<latexit 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<latexit 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  28. Overview • Sparse Linear Models • Sparse Estimation with Blasso

    • Non-super-resolution Regime • Super-resolution Regime
  29. Fisher-Rao Distance Fisher metric: <latexit 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<latexit 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<latexit 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<latexit 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Hx def. = [@'(x)][@'(x)]> 2 Rd⇥d <latexit 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<latexit 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  30. Fisher-Rao Distance Fisher metric: <latexit 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<latexit 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<latexit 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<latexit sha1_base64="6yIOx85vGmPIGpVqPjk65dRbuQQ=">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</latexit> Statistical interpretation: (|'(x)i |2/||'(x)||2)i ⇠ probability distribution. <latexit 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<latexit 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Hx def. = [@'(x)][@'(x)]> 2 Rd⇥d <latexit 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<latexit 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Fisher-Rao geodesic distance: <latexit 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dFR(x, x 0)2 def. = inf :x!x0 Z 1 0 hH (t) 0(t), 0(t)idt <latexit 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<latexit 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<latexit 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  31. Fisher-Rao Distance Fisher metric: <latexit 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<latexit 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<latexit 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<latexit sha1_base64="6yIOx85vGmPIGpVqPjk65dRbuQQ=">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</latexit> Statistical interpretation: (|'(x)i |2/||'(x)||2)i ⇠ probability distribution. <latexit 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<latexit 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Hx def. = [@'(x)][@'(x)]> 2 Rd⇥d <latexit 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<latexit 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Fisher-Rao geodesic distance: <latexit 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dFR(x, x 0)2 def. = inf :x!x0 Z 1 0 hH (t) 0(t), 0(t)idt <latexit 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<latexit 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<latexit 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dFR(x, x0) = 1||x x0|| <latexit 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<latexit 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<latexit sha1_base64="bWcvFxl/DTtomnoFaC6cQnZS9cI=">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</latexit> Deconvolution: X = Rd <latexit 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<latexit 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sha1_base64="pc8SAsN7XKd+k4XhYfRjr1XfX2o=">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</latexit> '(x) = h((· x)/s) <latexit 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  32. Fisher-Rao Distance Fisher metric: <latexit 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<latexit 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<latexit 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<latexit sha1_base64="6yIOx85vGmPIGpVqPjk65dRbuQQ=">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</latexit> Statistical interpretation: (|'(x)i |2/||'(x)||2)i ⇠ probability distribution. <latexit 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<latexit 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Hx def. = [@'(x)][@'(x)]> 2 Rd⇥d <latexit 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<latexit 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Fisher-Rao geodesic distance: <latexit 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dFR(x, x 0)2 def. = inf :x!x0 Z 1 0 hH (t) 0(t), 0(t)idt <latexit 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<latexit 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<latexit 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dFR(x, x0) = fc ||x x0|| <latexit 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<latexit 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<latexit 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Fourier coe cients: X = (R/Z)d <latexit 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<latexit 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<latexit sha1_base64="sTu0JnojxSUKNe/629PdimIg3Ww=">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</latexit> '(x) = (e2i⇡kx)fc k= fc <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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dFR(x, x0) = 1||x x0|| <latexit 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<latexit 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<latexit sha1_base64="bWcvFxl/DTtomnoFaC6cQnZS9cI=">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</latexit> Deconvolution: X = Rd <latexit 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<latexit 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sha1_base64="pc8SAsN7XKd+k4XhYfRjr1XfX2o=">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</latexit> '(x) = h((· x)/s) <latexit 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  33. Fisher-Rao Distance Fisher metric: <latexit 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<latexit 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<latexit 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<latexit sha1_base64="6yIOx85vGmPIGpVqPjk65dRbuQQ=">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</latexit> Statistical interpretation: (|'(x)i |2/||'(x)||2)i ⇠ probability distribution. <latexit 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<latexit 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Hx def. = [@'(x)][@'(x)]> 2 Rd⇥d <latexit 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<latexit 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Fisher-Rao geodesic distance: <latexit 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dFR(x, x 0)2 def. = inf :x!x0 Z 1 0 hH (t) 0(t), 0(t)idt <latexit 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<latexit 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<latexit 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dFR(x, x0) = fc ||x x0|| <latexit 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<latexit 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<latexit 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Fourier coe cients: X = (R/Z)d <latexit 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<latexit 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<latexit sha1_base64="sTu0JnojxSUKNe/629PdimIg3Ww=">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</latexit> '(x) = (e2i⇡kx)fc k= fc <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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dFR(x, x0) = | log(x/x0)| <latexit 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<latexit 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<latexit sha1_base64="KJVmwaWFwrPf55qWHtNYw5yt8is=">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</latexit> Laplace transform: X = R⇤ + <latexit 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<latexit 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<latexit sha1_base64="EJ1We4aVrjhS9LCFTWV8PtdfWUQ=">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</latexit> '(x) = e x· <latexit 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<latexit 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<latexit 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dFR(x, x0) = 1||x x0|| <latexit 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<latexit 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<latexit sha1_base64="bWcvFxl/DTtomnoFaC6cQnZS9cI=">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</latexit> Deconvolution: X = Rd <latexit 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<latexit 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sha1_base64="pc8SAsN7XKd+k4XhYfRjr1XfX2o=">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</latexit> '(x) = h((· x)/s) <latexit 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  34. Gaussian Mixtures x = (µ, ) 2 X = R

    ⇥ R+ <latexit 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<latexit 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<latexit 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<latexit sha1_base64="pFpLVhYssZkZ+t4dffb9/3PS/FU=">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</latexit> '(x) = 1 e (· µ)2 2 2 <latexit 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<latexit 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<latexit sha1_base64="/BT4VkLRHmogVsl6xepGj3yRs9M=">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</latexit> Hx = 1 2 Id2⇥2 <latexit 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<latexit 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<latexit sha1_base64="JcjP+v+mlmBmCNnV5cWAwnOcorY=">AABB6XictVzNchu5EYY3f2vnz5scc5lE62R3y3EkxVVJ1dZWrSzJkta0LZuU7N2lrRqSI3qsIYeeIWXZXD5CDrmlcs0j5JpUniNvkJzyCuluAAMMiZnGKI5RkjAgvu5GD9DoboDuTZI4n66v//PKe9/69ne++733r177/g9++KMfX//gJ8d5Osv60VE/TdLsaS/MoyQeR0fTeJpETydZFI56SfSkd7aNnz85j7I8Tsed6ZtJ9GwUDsfxadwPp9B0cv1X+ycXwWfd0yzszzcW824eP59vLhZB92BwMt8MutN4FOXB5uLk+tr6rXX6F6xWNlRlTah/h+kHwZHoioFIRV/MxEhEYiymUE9EKHIoX4sNsS4m0PZMzKEtg1pMn0diIa4Bdga9IugRQusZ/B7C09eqdQzPSDMndB+4JPCTATIQNwCTQr8M6sgtoM9nRBlbq2jPiSbK9gb+9hStEbROxQto5XC6py8OxzIVp+L3NIYYxjShFhxdX1GZkVZQ8sAa1RQoTKAN6wP4PIN6n5BazwFhcho76jakz/9FPbEVn/uq70z8m6S8ASUQbTX6tKAQinOiH9DbnMFnUp4EOA+BQqTGiLXXpOsRjX4M/efQ/gDKgmpaJz0oc2pd1CK3obiQ2yxyD4oLucciW1BcyBaLPITiQh4qJGIz0rkb34biwrdZzo+guJCPWORjKC7kYxZ5DMWFPGaRX0FxIb9ikXehuJB3WeQ9KC7kPRbZgeJCdljkERQX8ohF7kJxIXcVsnqlZlBSohMzq3IL6mUeaCkSaNli5btD1tGFveOxpvsVWH5V78BfN3bHQ6dRBXbXY96dVmD5mbcHNtKN5W3RPu0mLuw+iz2AGeDGHrDYL8TLCuwXHivtrALLr7UW9HNjeet7H57c2Pss9gHU3Fh+j3oILW7sQ48dY1KBPWSxj8SrCqyP1c8qsLzdb4NdcWP5faoD/d1YH2s6q8Dy9vQYPBg3lt+tnkCrG/uExT4VFxXYpyz2S7DubuyXHjvs2wqs3mOv0Q4yJH8kghVbRy0sViXWJkAtZPgnxd6SkG/cg3YOMywwQ8KMWMRegdjzRLQKRMtbrrywozn5uzyXdoFoeyJ6xd6EtSnbf1D0x1rigdgpEDtLiDqPFN+1Hss5eRe6hUNOi50Laz5jSgv7jbVIzYd6y6sRD0sIObdf0My/SdESRlCoqTpqL4o9XiIDeq5DvKboTY9S8+Bx08Iq2KgLFtVzoHos6o0D9YZFzRyoGYs6d6DOWZRZ+Tau6zEDjP7xXczpSc4A6SNXlwC8gi3YdfZhjQYwfw7BC3xMLQ/hb5tib67USYbRPO6TmOV4VrLEGdTmYg3aTVS4Q/F1QissAslkz4cqxscnzG3M1ZqTVnhR7ORBkTHxpxOTPMOCDnqLAa2nZnTuUcuCvDtZa4bfL9a9rjXD75LGF+TFy1oz/FRJP72E7B2F7VwC24bVNFHaN/WmNGT+RdLQ9Wu066LFxbc6UnMG6V00pH+g3szBJd7LNtWkfky9GY3cGl9eGl8TGkbPuaXnZlTQe5Jer64FjUcyVnGvqTeVIaVddKzkME9N3wz2Gag3o+vNaByCx7VNMffcqjedvZNiNKbejMaxkHnPBXnyut6MxpCepT5MvRkNzLaEKs439aaWHTUgY2dTb2rVx5QFxhyQnPOyxXhFGflJM0UtJv+gPltj+/yr+xjmbJ4XMUI9JePbVtPpFXtZvUTaX4jAqk0byoH+xczywco05mKTja+kDNPS/r5Kx+zxqPkWaDGA1S/PALiceQIS6pwEWu8EKG6wUVd5ZBq3yeJwlpwuobqqdcp6i4avzBqV206olYvLzGiNHrtkr3OaexPyCVukWU4Prco3XEWR01CrpCGeXhPdvVXrtaz9dRY3WUJMipnWpxMheZJWH6e6tN62dHxDnfJMocgzHzN/Mdt8qqwNxjwp2SKUpY6n3U/nkew23FdvCpPjlp8F9EbRXp2T1YjpRCpno1CdLZbe+JyeDe0jOpNDHpJGH95joKhMhDw1wyw65tMDsqi2veV4o750hk7Wc7K62h7Xo4cWeuhAN49xtmHHeAC1DsQMR/DU8YhyrhW6Sknjmfh1cTqa0husj+iTkoXUNKS9iUoWsi7KflGi8hrQOBtklO5PY5mOxndXKPFRv0seE7uWLf8NOrnV59shzfHq2VydiRkQ103iGtCqkae68mmZg5Rg7vxkk/zX+lEivyYc0YZyXJ9bnKVexnTiH1EEOyHPOKHVxq2Ocm87P7X8ieZ0KPTZOZ5mp2QhA7J/AexPKc3JgH7suwP6BF1ahIRspI/diQvvxuXrxOwcM35cLOStBjPfIrJlM+Kv6dqrK6e5KCMGuQ8slua21kmLfMGIuGbKupu1Xb/7INLck7BniaRo5spHxP9j+q1/9DxZW5kRqGF8A7myda73kVLMgjoKaZevt0G6ry3lh4UMz5XUZv8zMn1YkmyHIi6UB3frAXDu07PkhbMkI7nzlT5yH63L5iLlyZIecbSnFMVLuz9UOzDKfZN2yTVac12aJUOYBdMiitB9uSzyMt96XmXqfrTz/wt1o+uy1pBiIEwGV2qIy+9HFK3ZUiYwq+X8PaPV5NZ6ttSrns+Y5uLIWsvfQOvP4beWWz/70emVrMIdmgOSgnkyGpEtwUoPP153Srz0zNS0zLPhZ+ak7mW3XCa+ltbNxNjnjakc0qy5UFkLXb8MjZcWjZeeOuzQWaPRom7XluiEjS066rTSl18Tbp0GlGcsZd4j06jYQ0o7lvKjOmCp8jG+Rr1laa2ztEJYrfZpgL3mfZDutb68ur8pdvdA3CXfpk8emIxfBrRKY/K5dGt9pCYpIOfbyr7aq79LLci9RxYUKct7nLhi5KlTn8qikPSXamdLyc4bi6DvLb1WfbSN7VL9tyvIEa2JnNalRtymHpGS35YjWLJItyyfI6DMf0g+lfQ76mNmu7d5J0HJnzDxplxVhpeMFMakfy7zdrASvR5Y8WtAMeFMedc9oNX8DSMFidGZBLdnmdMbwl1OniRIj7ZH9nPVTslTvLEl0S2Sei4+87AxMuo1c92eW3rEemyfQE/Uunnrrh48v8SbI8fvMid6Ie1qI+WjzpeeL0crVLtc+blOD7MlvkYfM+pjRxYmyitjuuJTby5SomZcJMaHS7NRNJG/meRNZJanU76UdW9NuZxpkDbmBcVL3D1QRLi8u4+c3tzHzDh6K/R6hLWpyRaOEmbjUpUfsC0tZqWuruxDsvVq7W6UWDtR1U6hqdu7hbHf0kJGZP0SweVsZG9b9m4pSuGzMJJCX8gbvVXxoU3zUyj4OxCu6FBz9MkdtsG/3RLbYvcd3IZ4peoyoxlQC9qCwVLsHapxlnvU6+iVRd2m78PBn0cMuuakj2knbSq7pMxLblP3p/+arEAmIlZ607P5GGwu/EhWOTUZT0yWjR9NLPR3cZqORXPwGUmZiz8fea7BjeJU6O80NRuDps6PoMyhCQ99j8HvnZvezXnZnOr1tcrFl4fcBfSJi8bhyV91rGL6+ViozHoj754DWofTGup6t/hfx6H5GE7Neflyy+m7Zi893rrsF6mMLPrDzdeM4eYzm6s5+vNMi9EZb8nNT/p9QaM3lVqjeff00R81c0DzmguZB+Wlk3h7Fhl5fanguYBLhlT8R/zjCv9thFcFjSo5mlDS5xTV1HQPnpr+xqVrdPozH5kMnSqZytRMHNGmG7Hb4kDchZ/twgNsejtUfpdS/kWs+/uzA2g9Jeuhs+gyc9CltoiyH+YUbUDP5v5slcR4l1fe7e1AC56Ft6gV7/k+oP5417dTGlv1N0jkWr8vUjEoRSTLp3tmXfVgBOWTN5kD0t/zDeguvcxiyZtnI4+zRX1/almiOX3C3yzoVeJ7lpR9mqsTdVaPJwd4wz4s8kOB+A21hcrO457LcT6s5Hy4xDkn7ZQ5XFif1d/NquKybXEZFLmzc9UvpTjbnOfV50Z3KrnIO+j1+GENfmhJ2Sbtn1EknIn6bN6shuZMyWSfsI6FzkRKPWCcGRbvuz6yPa/hde4x/nuV6HuWpHsgS4/y3wGdsGVEL1G62SXp5U3H+kzqfo206nuUJ9fXNpb/L4PVyvHmrY31WxuPbq99fkf9Pwfvi5+JX4iPYI3/TnwO1A7FEXD4g/ir+Jv4+9bZ1h+3/rT1Z9n1vSsK81NR+rf1l/8CaJyvyA==</latexit> dFR(x, x0) = arcsinh ||x x0|| p 0 <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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  35. Exact Support Recovery Input measure: m0 = P i ai

    xi . <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="ooBrb91hD76ixeX7Opvhtv48d/8=">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</latexit> Theorem: there exists such that <latexit 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<latexit 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sha1_base64="PTN1DUCaapZR3FVie4BE16FhL4c=">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</latexit> ⌘V is non-degenerate. <latexit 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<latexit 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<latexit 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Remark: additional smoothness/decay assumption required. <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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[Poon, Keriven, P, 2018] <latexit 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<latexit 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<latexit 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Exclusion balls: <latexit 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<latexit 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={x0 ; dFR(x, x0) 6 } <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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x1 <latexit 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x2 <latexit 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⇠ f 1 c <latexit 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<latexit sha1_base64="dB9UhR1Y30UW+TR7Xr81NkAKMTc=">AABBzXictVzbchu5EYU3t7VzWW/ymJdJtE55U15HclyVVG2lamVJlrWmLdqkZO8ubRcvI5r2kENzSPlCK6/5hLwmX5LvyB8kT/mF9AUYYEjMNEZxjJKEAXG6Gz1Ao7sBujdNRtl8c/OfFz763vd/8MMffXzx0o9/8tOffXL5058fZ+li1o+P+mmSzh73ulmcjCbx0Xw0T+LH01ncHfeS+FHv5Q5+/ug0nmWjdNKev53GT8bd4WR0Mup359D0pJONxtHJs/7T5RdbZ88ub2xe36R/0XplS1c2lP7XTD+NjlRHDVSq+mqhxipWEzWHeqK6KoPyndpSm2oKbU/UEtpmUBvR57E6U5cAu4BeMfToQutL+D2Ep+906wSekWZG6D5wSeBnBshIXQFMCv1mUEduEX2+IMrYWkZ7STRRtrfwt6dpjaF1rp5Dq4QzPUNxOJa5OlF/pDGMYExTasHR9TWVBWkFJY+cUc2BwhTasD6Az2dQ7xPS6DkiTEZjR9126fN/UU9sxee+7rtQ/yYpr0CJVEuPPs0pdNUp0Y/obS7gM5YnAc5DoBDrMWLtNel6TKOfQP8ltN+HckY1o5MelCW1nlUid6D4kDsich+KD7kvIhtQfMiGiGxC8SGbGonYGencj29B8eFbIucHUHzIByLyIRQf8qGIPIbiQx6LyG+h+JDfisjbUHzI2yLyLhQf8q6IbEPxIdsi8giKD3kkIveg+JB7Glm+UmdQUqIzElblNtSLPNBSJNCyLcp3i6yjD3srYE33S7Dyqt6Fv37sboBO4xLsXsC8OynByjNvH2ykHyvboju0m/iwd0TsAcwAP/ZAxH6tXpRgvw5YaS9LsPJaa0A/P1a2vvfgyY+9J2LvQ82PlfeoQ2jxYw8DdoxpCbYpYh+oVyXYEKs/K8HKdr8FdsWPlfepNvT3Y0Os6aIEK9vTY/Bg/Fh5t3oErX7sIxH7WL0pwT4Wsd+AdfdjvwnYYd+VYM0ee4l2kCH5IzGs2Cpq3XxVYm0K1LoC/yTfWxLyjXvQLmGGOWZImLGI2M8R+4GIRo5oBMuV5XY0I39X5tLKEa1ARC/fm7A2F/sP8v5YSwIQuzlidwVR5ZHiuzZjOSXvwrRIyHm+c2EtZExpbr+xFuv5UG15DeKwgOC5/Zxm/jWKljCCQk1VUXue7/GMjOi5CvGaojczSsNDxs1zq+Ci3oiongfVE1FvPai3ImrhQS1E1KkHdSqi7Mp3cZ2AGWD1j+9iSU88A9hHLi8ReAXbsOvcgTUawfxpghf4kFoO4W+LYm+pVEmG0Tzuk5jleFKwxDOoLdUGtNuocJfi64RWWAyScc9DHePjE+Y2lnrNsRU+y3fyKM+YhNMZkTzDnA56ixGtp3p07lLLGXl3XKuHv5Ove1Orh98jjZ+RF8+1evi5ln5+DtnbGts+B7YFq2mqtW/rdWlw/oVpmPol2nXR4uJbHes5g/Te1KR/oN/MwTneyw7VWD+2Xo9G5owvK4yvDg2r58zRcz0q6D2x12tqUe2RTHTca+t1ZUhpF51oOexT3TeDfQb6zZh6PRpN8Lh2KOZeOvW6s3eaj8bW69E4Vpz3PCNP3tTr0RjSM+vD1uvRwGxLV8f5tl7XsqMGOHa29bpWfUJZYMwB8ZznFusVzchPWmhqI/IPqrM1rs+/vo9hzuZpHiNUU7K+bTmdXr6XVUtk/IUYrNq8phzoXywcH6xIY6luiPEVyzAv7O/rdOwej5pvgBYjWP18BiDlzBOQ0OQk0HonQHFLjLqKIzO4GyIOZ8nJCqqjW+eit2j5ctao2PaMWqW4zI7W6rFD9jqjuTcln7BBmpX00Ch9w2UUJQ01ChqS6dXR3Tu9Xova3xRx0xXENJ9pfToR4pO06jjVp/WWo+Mr+pRnDoXPfOz8xWzzibY2GPOkZItQliqebj+TR3LbcF+9pmyOmz+L6I2ivTolqzGiE6lMjEJNtpi98SU9W9pHdCaHPJhGH95jpKlMFZ+aYRYd8+kRWVTX3kq8UV8mQ8f1jKyuscfV6KGDHnrQ9WOcHdgx7kOtDTHDETy1A6KcS7muUtL4TH2Rn46m9AarI/qkYCENDbY3ccFCVkXZzwtUXgMaZwNH6eE0VukYfGeNkhz1++SxsWvR8l+hk1tzvt2lOV4+m8szMQPieoO4RrRq+FSXn1Y5sARL7yc3yH+tHiXyq8MRbajE9anDmfUyoRP/mCLYKXnGCa02aXUUe7v5qdVPDKemMmfneJqdkoWMyP5FsD+lNCcj+nHvDpgTdLYICdnIELszyr0bn68zEueY9eNGim812PkWky1bEH9D111dGc1Fjhh4HzhbmdtGJw3yBWPiOtPW3a7t6t0HkfaehDtLmKKdK1eJ/+f02/yYebKxNiNQw/gGMm3rfO8jpZgFddSlXb7aBpm+rpSf5TI81VLb/c/K9FlBsl2KuFAe3K0HwLlPz8wLZ8mM5M7W+vA+WpXNRcrTFT3iaE8oime7P9Q7MMp9jXbJDVpzHZolQ5gF8zyKMH2lLPIq32peRephtLP/C3Wr66LWkGKkbAaXNSTl92OK1lwpE5jVPH9f0mrya3220quaz4Tm4thZy++h9Vfw28htnsPo9ApW4RbNAaZgn6xGuCVa6xHG61aBl5mZhpZ9tvzsnDS93JbzxNds3WyMfVqbSpNmzRudtTD189B44dB4EajDNp01Wi2admOJnomxRVufVobyq8OtXYPyQqQse2QGNQqQ0o2lwqgORKpyjG9Q70RamyKtLqxW9zTAXfMhSP9aX13d7/PdPVK3ybfpkwfG8cuAVumIfC7TWh2pMQXkfFPbV3f1d6gFuffIgiJlvseJK4ZPnfpUznJJf6N3tpTsvLUI5t7Sa93H2NgO1X+/hhzTmshoXRrETeoRa/ldOaIVi3Td8Tkiyvx3yadiv6M6ZnZ723cSFfwJG2/yqrK8OFKYkP6lzNvBWvR64MSvEcWEC+1d94BW/TeMFBhjMgl+zzKjN4S7HJ8ksEfbI/u5bqf4FG/iSHSdpF6qPwXYGI567Vx355YZsRnbb6Enat2+dV8PmV8SzFHid54TvS7tamPtoy5Xns9Hq6t3ueJzlR4WK3ytPhbUx40sbJRXxHTUl8FcWKJ6XBgTwqXeKOrIX0/yOjLz6VQoZdPbUC5mGtjGPKd4SboHigifd3fV6819Loyjt0avR1iXGrdIlDAbl+r8gGtpMSt1cW0f4taLlbtR4uxEZTuFoe7uFtZ+s4WMyfolSsrZcG9X9k4hSpGzMEyhr/hGb1l86NL8Egr+jpQvOjQcQ3KHLfBvt9WO2vsAtyFe6TpnNCNqQVswWIm9u3qcxR7VOnrlUHfph3AI5zECXUvSj2gnrSs7U5Yld6mH039NVmCmYlF627P+GFwu8kjWOdUZz4gsmzyakTLfxak7FsMhZCRFLuF8+FxDGsWJMt9pqjcGQ10eQZFDHR7mHkPYO7e96/NyOVXra51LKA/eBcyJi8HhyV95rGL7hViomfNGPjwHtA4nFdTNbvG/jsPwsZzq8wrlltF3zV4EvHXuF+uMLPrD9deM5RYym8s5hvNM89FZb8nPj/2+qNabSp3RfHj66I/aOWB4LRXnQWXpGO/OIitvKBU8F/DJkKr/qH9ckL+N8CqnUSZHHUrmnKKcmukhUzPfuPSNznwWIpOlUyZTkZqNI1p0I3ZHHajb8LOTe4B1b4fydyn5L2L9358dQOsJWQ+TRefMQYfaYsp+2FO0AT3b+7NlEuNdXr7b24YWPAtvUCve871P/fGub7swtvJvkPBav6dSNShEJKune3Zd9WAExZM3zgGZ7/lGdJees1h882wccLZo7k+tSrSkT+SbBb1SfM+Rsk9zdarP6vHkAG/Yd/P8UKR+R21dbedxz5U4N0s5N1c4Z6SdIoc3zmfVd7PKuOw4XAZ57uxU90spzrbnedW50d1SLnwHvRo/rMAPHSlbpP2XFAnPVHU2b1FBc6Flck9YJ8pkIlkPGGd28/ddHdmeVvA6DRj/3VL0XUfSfZClR/nviE7YZkQv0brZI+n5pmN1JvVOhbT6e5TPLm9srf5fBuuV4xvXtzavbz24ufHVLf3/HHysfql+ra7CGv+D+gqoNdUR2du/qr+pv28fbi+232//mbt+dEFjfqEK/7b/8l+kgKU9</latexit> '(x) = (e2i⇡kx)fc k= fc <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="0gSeVDpv/E0yvhLCYTpyOva6aTc=">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</latexit> '(x) = h((· x)/s) <latexit 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<latexit 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'(x) = 1 e (· µ)2 2 2 <latexit 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<latexit 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<latexit 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  36. Exact Support Recovery Input measure: m0 = P i ai

    xi . <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="ooBrb91hD76ixeX7Opvhtv48d/8=">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</latexit> Theorem: there exists such that <latexit 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<latexit 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sha1_base64="PTN1DUCaapZR3FVie4BE16FhL4c=">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</latexit> ⌘V is non-degenerate. <latexit 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<latexit 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<latexit 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Remark: additional smoothness/decay assumption required. <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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[Poon, Keriven, P, 2018] <latexit 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<latexit 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<latexit 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Exclusion balls: <latexit 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<latexit 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={x0 ; dFR(x, x0) 6 } <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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x1 <latexit 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x2 <latexit 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⇠ f 1 c <latexit 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<latexit sha1_base64="dB9UhR1Y30UW+TR7Xr81NkAKMTc=">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</latexit> '(x) = (e2i⇡kx)fc k= fc <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="0gSeVDpv/E0yvhLCYTpyOva6aTc=">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</latexit> '(x) = h((· x)/s) <latexit 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<latexit 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'(x) = 1 e (· µ)2 2 2 <latexit 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<latexit 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Remark: extends to randomized (compressed sensing) measurements. <latexit 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<latexit 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Remark: for Fourier (or convolution), independent of fc (or s). <latexit 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<latexit 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<latexit 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<latexit 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  37. Optimal Transport Approximate Recovery “Unbalanced” Fisher-Rao Optimal Transport: <latexit sha1_base64="ANX5T3t4paZnI2xBwuWOoZRhBGc=">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</latexit>

    <latexit 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<latexit 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<latexit 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W2 2 (m1, m2) def. = inf ⇡2M+(X2) Z X2 d2 FR (x1, x2)d⇡(x1, x2) + |⇡1 m1 |(X) + |⇡2 m2 |(X) <latexit 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<latexit 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<latexit 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Example: Gaussian mixtures. <latexit 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<latexit 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<latexit sha1_base64="/BT4VkLRHmogVsl6xepGj3yRs9M=">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</latexit> '(x) = 1 e (· µ)2 2 2 <latexit 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<latexit 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⇡ <latexit 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  38. Optimal Transport Approximate Recovery “Unbalanced” Fisher-Rao Optimal Transport: <latexit sha1_base64="ANX5T3t4paZnI2xBwuWOoZRhBGc=">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</latexit>

    <latexit 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<latexit 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<latexit 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W2 2 (m1, m2) def. = inf ⇡2M+(X2) Z X2 d2 FR (x1, x2)d⇡(x1, x2) + |⇡1 m1 |(X) + |⇡2 m2 |(X) <latexit 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<latexit 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<latexit 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Example: Gaussian mixtures. <latexit 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<latexit 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<latexit sha1_base64="/BT4VkLRHmogVsl6xepGj3yRs9M=">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</latexit> '(x) = 1 e (· µ)2 2 2 <latexit 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<latexit 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⇡ <latexit 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|⌘(x)| 6 1 <latexit 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sha1_base64="Gt+ssAS/RrevEzTu2WMnRp8UNZE=">AABB4HictVxfcxPJER8u/w7yj0seU5XaxEcKUoTYhKqk6ipVZ2xjfBgwSDbcncClldZCsNYKrWQMwm95yFsqr/kIeU3e8znyDZKnfIX0n5mdWWl2e9YhTNmeHc2vu6d3pqe7Z0Q8Tof5dHX1nxc++sY3v/Xt73x88dJ3v/f9H/zw8ic/Osiz2aSX7PeyNJs8jbt5kg5Hyf50OE2Tp+NJ0j2O0+RJ/GoDP39ykkzyYTZqT9+Ok2fH3cFoeDTsdafQdHj5p+/nHaIyj9NZctZJpt2zq6fX3kedNHkdrR1eXlm9sUr/ouXKmq6sKP1vL/sk2lcd1VeZ6qmZOlaJGqkp1FPVVTmUr9WaWlVjaHum5tA2gdqQPk/UmboE2Bn0SqBHF1pfwe8BPH2tW0fwjDRzQveASwo/E0BG6gpgMug3gTpyi+jzGVHG1irac6KJsr2Fv7GmdQytU/UCWiWc6RmKw7FM1ZH6HY1hCGMaUwuOrqepzEgrKHnkjGoKFMbQhvU+fD6Beo+QRs8RYXIaO+q2S5//i3piKz73dN+Z+jdJeQVKpFp69FlBoatOiH5Eb3MGn7E8KXAeAIVEjxFrb0jXxzT6EfSfQ/sDKGdUMzqJocyp9awWuQHFh9wQkdtQfMhtEbkLxYfcFZF7UHzIPY1E7IR07se3oPjwLZHzIyg+5CMR+RiKD/lYRB5A8SEPRORXUHzIr0TkHSg+5B0ReQ+KD3lPRLah+JBtEbkPxYfcF5FbUHzILY2sXqkTKBnRGQqrch3qZR5oKVJoWRflu03W0Ye9HbCmexVYeVVvwl8/djNAp0kFditg3h1VYOWZtw020o+VbdFd2k182LsidgdmgB+7I2K/UC8rsF8ErLRXFVh5re1CPz9Wtr734cmPvS9iH0DNj5X3qIfQ4sc+DNgxxhXYPRH7SL2uwIZY/UkFVrb7LbArfqy8T7Whvx8bYk1nFVjZnh6AB+PHyrvVE2j1Y5+I2KfqtAL7VMR+Cdbdj/0yYId9V4E1e+wl2kEG5I8ksGLrqHWLVYm1MVDrCvzTYm9JyTeOoV3CDArMgDDHImK7QGwHInYLxG6wXHlhR3Pyd2UurQLRCkTExd6EtanYv1/0x1oagNgsEJsLiDqPFN+1GcsJeRemRUJOi50LayFjygr7jbVEz4d6y2sQD0sIntsvaOZfp2gJIyjUVB21F8Uez8iInusQbyh6M6M0PGTctLAKLupURMUeVCyi3npQb0XUzIOaiagTD+pERNmV7+I6ATPA6h/fxZyeeAawj1xdIvAK1mHXuQtrNIL5swde4GNqeQh/WxR7S6VOMozmcZ/ELMezkiWeQG2uVqDdRoWbFF+ntMISkIx7PtQxPj5hbmOu1xxb4bNiJ4+KjEk4nSHJMyjooLcY0XpqRucetZyRd8e1Zvi7xbo3tWb4LdL4GXnxXGuGn2rpp+eQva2x7XNgW7Caxlr7tt6UBudfmIapX6JdFy0uvtVjPWeQ3mlD+jv6zeyc471sUI31Y+vNaOTO+PLS+JrQsHrOHT03o4LeE3u9phY1HslIx7223lSGjHbRkZbDPjV9M9inr9+MqTejsQce1wbF3HOn3nT2jovR2HozGgeK855n5MmbejMaA3pmfdh6MxqYbenqON/Wm1p21ADHzrbe1KqPKAuMOSCe89xivaIJ+UkzTW1I/kF9tsb1+Zf3MczZPC9ihHpK1retphMXe1m9RMZfSMCqTRvKgf7FzPHByjTm6qYYX7EM09L+vkzH7vGo+V3QYgSrn88ApJx5ChKanARa7xQorolRV3lkBndTxOEsOVpAdXTrVPQWLV/OGpXbDqlVisvsaK0eO2Svc5p7Y/IJd0mzkh52K99wFUVJQ7slDcn0mujunV6vZe2virjxAmJczLQenQjxSVp9nOrTesvR8RV9yjOFwmc+dv5itvlIWxuMeTKyRShLHU+3n8kjuW24r15XNsfNn0X0RtFenZDVGNKJVC5GoSZbzN74nJ4t7X06k0MeTKMH7zHSVMaKT80wi4759IgsqmtvJd6oL5Oh43pOVtfY43r0wEEPPOjmMc4G7BgPoNaGmGEfntoBUc6lQlcZaXyiflWcjmb0Busj+rRkIQ0NtjdJyULWRdkvSlTeABpnA0fp4TQW6Rh8Z4mSHPX75LGxa9nyX6GTW3O+3aU5Xj2bqzMxfeJ6k7hGtGr4VJefFjmwBHPvJzfJf60fJfJrwhFtqMT1ucOZ9TKiE/+EItgxecYprTZpdZR7u/mpxU8Mpz1lzs7xNDsjCxmR/Ytgf8poTkb0494dMCfobBFSspEhdmdYeDc+X2cozjHrxw0V32qw8y0hWzYj/oauu7pymoscMfA+cLYwt41OdskXTIjrRFt3u7brdx9E2nsS7ixhinauXCX+1+i3+THzZGVpRqCG8Q3k2tb53kdGMQvqqEu7fL0NMn1dKT8tZHiupbb7n5Xp05JkmxRxoTy4W/eBc4+emRfOkgnJnS/14X20LpuLlMcLesTRHlEUz3Z/oHdglPs67ZIrtOY6NEsGMAumRRRh+kpZ5EW+9bzK1MNo5/8X6lbXZa0hxUjZDC5rSMrvJxStuVKmMKt5/r6i1eTX+mShVz2fEc3FY2ctv4fWn8FvI7d5DqMTl6zCbZoDTME+WY1wS7TUI4zX7RIvMzMNLfts+dk5aXq5LeeJr9m62Rj7pDGVPZo1pzprYernofHSofEyUIdtOmu0WjTtxhIdirFFW59WhvJrwq3dgPJMpCx7ZAY1DJDSjaXCqPZFqnKMb1DvRFqrIq0urFb3NMBd8yFI/1pfXN3vi909UnfIt+mRB8bxS59W6ZB8LtNaH6kxBeR8S9tXd/V3qAW5x2RBkTLf48QVw6dOPSpnhaS/0DtbRnbeWgRzb+mN7mNsbIfqv1lCHtOayGldGsQt6pFo+V05ogWLdMPxOSLK/HfJp2K/oz5mdnvbdxKV/Akbb/Kqsrw4UhiR/qXM285S9LrjxK8RxYQz7V3HQKv5G0YKjDGZBL9nmdMbwl2OTxLYo43Jfi7bKT7FGzkS3SCp5+r3ATaGo1471925ZUZsxvZL6Ilat2/d10PmlwZzlPid50SvS7vasfZR5wvP56PV1btc+blOD7MFvlYfM+rjRhY2yitjOuqzYC4sUTMujAnh0mwUTeRvJnkTmfl0KpSy6W0olzMNbGNeULwk3QNFhM+7u+r15q4J44iX6MWEdalxi0QJs3GZzg+4lhazUheX9iFuvVi7G6XOTlS1Uxjq7m5h7TdbyISsX6qknA33dmXvlKIUOQvDFHqKb/RWxYcuzc+g4O9I+aJDwzEkd9gC/3ZdbaitD3Ab4rWuc0Yzoha0Bf2F2Lurx1nuUa+j1w51l34Ih3AeQ9C1JP2QdtKmsjNlWXKXejj9N2QFJioRpbc9m4/B5SKPZJlTk/EMybLJoxkq812cpmMxHEJGUuYSzofPNaRRHCnznaZmYzDU5RGUOTThYe4xhL1z27s5L5dTvb6WuYTy4F3AnLgYHJ78Vccqtl+IhZo4b+TDc0DrcFRD3ewW/+s4DB/LqTmvUG45fdfsZcBb536JzsiiP9x8zVhuIbO5mmM4z6wYnfWW/PzY74savanMGc2Hp4/+qJ0DhtdccR5Ulo7x7iyy8oZSwXMBnwyZ+o/6xwX52wivCxpVcjShZM4pqqmZHjI1841L3+jMZyEyWTpVMpWp2TiiRTdiN9SOugM/G4UH2PR2KH+Xkv8i1v/92T60HpH1MFl0zhx0qC2h7Ic9RevTs70/WyUx3uXlu71taMGz8F1qxXu+D6g/3vVtl8ZW/Q0SXuv3Vab6pYhk8XTPrqsYRlA+eeMckPmeb0R36TmLxTfPjgPOFs39qUWJ5vSJfLMgrsTHjpQ9mqtjfVaPJwd4w75b5Ici9Wtq62o7j3uuxHmvkvPeAuectFPmcOp8Vn83q4rLhsOlX+TOTnS/jOJse55XnxvdrOTCd9Dr8YMa/MCRskXaf0WR8ETVZ/NmNTRnWib3hHWkTCaS9YBxZrd43/WR7UkNr5OA8d+rRN9zJN0GWWLKf0d0wjYheqnWzRZJzzcd6zOpd2uk1d+jPLy8srb4fxksVw5u3lhbvbH26NbK57f1/3PwsfqJ+rm6Cmv8t+pzoLan9oHDH9Rf1d/U39fj9T+u/2n9z9z1owsa82NV+rf+l/8Ck56s8A==</latexit> ⌘(xi) = sign(ai) <latexit 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Dual certificate: ⌘ 2 Im( ⇤), <latexit 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  39. Optimal Transport Approximate Recovery “Unbalanced” Fisher-Rao Optimal Transport: <latexit sha1_base64="ANX5T3t4paZnI2xBwuWOoZRhBGc=">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</latexit>

    <latexit 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<latexit 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<latexit 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W2 2 (m1, m2) def. = inf ⇡2M+(X2) Z X2 d2 FR (x1, x2)d⇡(x1, x2) + |⇡1 m1 |(X) + |⇡2 m2 |(X) <latexit 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<latexit 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<latexit 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Example: Gaussian mixtures. <latexit 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<latexit 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<latexit sha1_base64="/BT4VkLRHmogVsl6xepGj3yRs9M=">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</latexit> '(x) = 1 e (· µ)2 2 2 <latexit 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<latexit 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⇡ <latexit 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|⌘(x)| 6 1 <latexit 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sha1_base64="Gt+ssAS/RrevEzTu2WMnRp8UNZE=">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</latexit> ⌘(xi) = sign(ai) <latexit 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Dual certificate: ⌘ 2 Im( ⇤), <latexit 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<latexit sha1_base64="pkyN7IHXtiHSYH+1cpE5jzPOx/Q=">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</latexit> Theorem: [Poon, Keriven, Peyr´ e 2018] <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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If 9⌘ non-degenerate certificate <latexit 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<latexit 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sha1_base64="y3yV6E1nXcwTSBRrsn92xaCG24I=">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</latexit> then for any solution m of (P ( m0 + w)) <latexit 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<latexit 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! golfing scheme construction of ⌘. <latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="t8QDfkMqewIjhpOpxpBBGYoGTQQ=">AABCBHictVzNchu5EYY3f2vnz5scc5lEcsqbchRJ2aqkaitVqz/LWsu2bFKyd5e2akgOqbGHHHqGlGxzdc2z5JBbKtfkDXJN8gbJKa+Q7gYwwJCYaYziGCUJA+LrbvQAje4G6O4kifPp+vo/r33wjW9+69vf+fD6je9+7/s/+OHNj350kqezrBcd99IkzZ51wzxK4nF0PI2nSfRskkXhqJtET7uvdvDzp+dRlsfpuD19O4mej8LhOB7EvXAKTac3t1c7WTw8m4ZZll6sBsM0GcTjYZD3zqJRFPTScT7NZj3sG6SDYHXeIZbzbjKLLjvRNLxcXTu9ubK+tk7/guXKhqqsCPXvKP0oOBYd0Rep6ImZGIlIjMUU6okIRQ7lK7Eh1sUE2p6LObRlUIvp80hcihuAnUGvCHqE0PoKfg/h6SvVOoZnpJkTugdcEvjJABmIW4BJoV8GdeQW0OczooytVbTnRBNlewt/u4rWCFqn4gxaOZzu6YvDsUzFQPyWxhDDmCbUgqPrKSoz0gpKHlijmgKFCbRhvQ+fZ1DvEVLrOSBMTmNH3Yb0+b+oJ7bic0/1nYl/k5S3oASipUafFhRCcU70A3qbM/hMypMA5yFQiNQYsXZBuh7R6MfQfw7tD6FcUk3rpAtlTq2XtcgdKC7kDovch+JC7rPIQygu5CGLPILiQh4pJGIz0rkb34LiwrdYzo+huJCPWeQTKC7kExZ5AsWFPGGRX0JxIb9kkXehuJB3WeR9KC7kfRbZhuJCtlnkMRQX8phF7kFxIfcUsnqlZlBSohMzq3IL6mUeaCkSaNli5dsm6+jCbnus6V4Fll/Vu/DXjd310GlUgd3zmHeDCiw/8/bBRrqxvC26R7uJC3uPxR7ADHBjD1js5+JlBfZzj5X2qgLLr7VD6OfG8tb3ATy5sQ9Y7EOoubH8HvUIWtzYRx47xqQCe8RiH4vXFVgfq59VYHm73wK74sby+1Qb+ruxPtZ0VoHl7ekJeDBuLL9bPYVWN/Ypi30m3lRgn7HYL8C6u7FfeOyw7yqweo+9QTvIkPyRCFZsHbWwWJVYmwC1kOGfFHtLQr5xF9o5zLDADAkzYhH7BWLfE3FYIA695coLO5qTv8tzaRWIlieiW+xNWJuy/ftFf6wlHojdArG7gKjzSPFd67Gck3ehWzjktNi5sOYzprSw31iL1Hyot7wa8aiEkHP7jGb+HYqWMIJCTdVROyv2eIkM6LkOcUHRmx6l5sHjpoVVsFFvWFTXgeqyqLcO1FsWNXOgZizq3IE6Z1Fm5du4jscMMPrHdzGnJzkDpI9cXQLwCrZg17kHazSA+XMEXuATankEf1sUe3OlTjKM5nGfxCzH85IlzqA2FyvQbqLCXYqvE1phEUgmez5SMT4+YW5jrtactMKXxU4eFBkTfzoxyTMs6KC3GNB6akbnPrVckncna83w94p1r2vN8Huk8Uvy4mWtGX6qpJ9eQfa2wravgG3Bapoo7Zt6Uxoy/yJp6PoN2nXR4uJbHak5g/TeNKR/oN7MwRXeyw7VpH5MvRmN3BpfXhpfExpGz7ml52ZU0HuSXq+uBY1HMlZxr6k3lSGlXXSs5DBPTd8M9umrN6PrzWgcgce1QzH33Ko3nb2TYjSm3ozGiZB5z0vy5HW9GY0hPUt9mHozGphtCVWcb+pNLTtqQMbOpt7Uqo8pC4w5IDnnZYvxijLyk2aKWkz+QX22xvb5l/cxzNm8KGKEekrGt62m0y32snqJtL8QgVWbNpQD/YuZ5YOVaczFJhtfSRmmpf19mY7Z41Hzh6DFAFa/PAPgcuYJSKhzEmi9E6C4wUZd5ZFp3CaLw1kyWEB1VOuU9RYNX5k1KredUisXl5nRGj12yF7nNPcm5BMekmY5PRxWvuEqipyGDksa4uk10d07tV7L2l9ncZMFxKSYaT06EZInafVxqkvrLUvHt9QpzxSKPPMx8xezzQNlbTDmSckWoSx1PO1+Oo9kt+G+ekeYHLf8LKA3ivbqnKxGTCdSORuF6myx9Mbn9GxoH9OZHPKQNHrwHgNFZSLkqRlm0TGfHpBFte0txxv1pTN0sp6T1dX2uB49tNBDB7p5jLMDO8ZDqLUhZjiGp7ZHlHOj0FVKGs/EL4vT0ZTeYH1En5QspKYh7U1UspB1UfZZicoFoHE2yCjdn8YiHY3vLFHio36XPCZ2LVv+W3Ryq8+3Q5rj1bO5OhPTJ66bxDWgVSNPdeXTIgcpwdz5ySb5r/WjRH5NOKIN5bi+sDhLvYzpxD+iCHZCnnFCq41bHeXedn5q8RPN6Ujos3M8zU7JQgZk/wLYn1KakwH92HcH9Am6tAgJ2UgfuxMX3o3L14nZOWb8uFjIWw1mvkVky2bEX9O1V1dOc1FGDHIfuFyY21onh+QLRsQ1U9bdrO363QeR5p6EPUskRTNXbhP/j+m3/tHzZGVpRqCG8Q3kyta53kdKMQvqKKRdvt4G6b62lKuFDC+U1Gb/MzKtliTbpYgL5cHdug+ce/QseeEsyUjufKmP3EfrsrlIebKgRxztgKJ4afeHagdGue/QLrlCa65Ds2QIs2BaRBG6L5dFXuRbz6tM3Y92/n+hbnRd1hpSDITJ4EoNcfn9iKI1W8oEZrWcv69oNbm1ni30quczprk4stby19D6U/it5dbPfnS6JauwTXNAUjBPRiOyJVjq4cdru8RLz0xNyzwbfmZO6l52y1Xia2ndTIx93pjKEc2aNyproetXofHSovHSU4dtOms0WtTt2hKdsrFFW51W+vJrwq3dgPKMpcx7ZBoVe0hpx1J+VPssVT7G16h3LK11llYIq9U+DbDXvA/SvdYXV/fXxe4eiLvk2/TIA5PxS59WaUw+l26tj9QkBeT8ibKv9urvUAty75IFRcryHieuGHnq1KNyWUj6c7WzpWTnjUXQ95YuVB9tYztU//USckRrIqd1qRGfUI9IyW/LESxYpDXL5wgo8x+STyX9jvqY2e5t3klQ8idMvClXleElI4Ux6Z/LvB0sRa8HVvwaUEw4U951F2g1f8NIQWJ0JsHtWeb0hnCXkycJ0qPtkv1ctlPyFG9sSbRGUs/F7zxsjIx6zVy355YesR7bL6Anat28dVcPnl/izZHjd5UTvZB2tZHyUecLz1ejFapdrvxcp4fZAl+jjxn1sSMLE+WVMR3xqTcXKVEzLhLjw6XZKJrI30zyJjLL0ylfyrq3plzONEgbc0bxEncPFBEu7+6205v7mBlHd4lel7A2NdnCUcJsXKryA7alxazU9aV9SLZer92NEmsnqtopNHV7tzD2W1rIiKxfIricjexty94pRSl8FkZS6Al5o7cqPrRpfgoFfwfCFR1qjj65wxb4t1tiR+y9h9sQr1VdZjQDakFb0F+IvUM1znKPeh29tqjb9H04+POIQdec9DHtpE1ll5R5yW3q/vQvyApkImKlNz2bj8Hmwo9kmVOT8cRk2fjRxEJ/F6fpWDQHn5GUufjzkeca3CgGQn+nqdkYNHV+BGUOTXjoewx+79z0bs7L5lSvr2UuvjzkLqBPXDQOT/6qYxXTz8dCZdYbef8c0DoMaqjr3eJ/HYfmYzg15+XLLafvmr30eOuyX6QysugPN18zhpvPbK7m6M8zLUZnvCU3P+n3BY3eVGqN5v3TR3/UzAHNay5kHpSXTuLtWWTk9aWC5wIuGVLxH/HXa/y3EV4XNKrkaEJJn1NUU9M9eGr6G5eu0enPfGQydKpkKlMzcUSLbsTuiANxF352Cg+w6e1Q+V1K+Rex7u/P9qF1QNZDZ9Fl5qBDbRFlP8wpWp+ezf3ZKonxLq+829uGFjwLP6RWvOf7kPrjXd92aWzV3yCRa/2BSEW/FJEsnu6ZddWFEZRP3mQOSH/PN6C79DKLJW+ejTzOFvX9qUWJ5vQJf7OgW4nvWlL2aK5O1Fk9nhzgDfuwyA8F4lfUFio7j3sux/mokvPRAuectFPm8Mb6rP5uVhWXHYtLv8idnat+KcXZ5jyvPje6W8lF3kGvxw9r8ENLyhZp/xVFwpmoz+bNamjOlEz2CetY6Eyk1APGmWHxvusj2/MaXuce479fib5vSboPsnQp/x3QCVtG9BKlmz2SXt50rM+k3quRVn2P8vTmysbi/2WwXDnZXNtYX9t4vLny2bb6fw4+FD8RPxO3YY3/RnwG1I7EMXD4g/ib+Lv4x9bvt/649aetP8uuH1xTmB+L0r+tv/wXdNW75A==</latexit> W 2 2 (m , m0) = O(||w||) when ⇠ ||w|| <latexit 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<latexit 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<latexit 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<latexit 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  40. Overview • Sparse Linear Models • Sparse Estimation with Blasso

    • Non-super-resolution Regime • Super-resolution Regime
  41. Recovery of Positive Measures Theorem: let and [de Castro et

    al. 2011] ! m0 is recovered when there is no noise. ⌘S(t) = 1 ⇢ QN i=1 sin(⇡(t xi))2 -1 1 -1 1 -1 1 -1 1 ⌘S ⌘S ⌘S ⌘S Input measure: m0 = ma,x where a 2 RN + . '(x) = (ei`x)fc `= fc for N 6 fc and ⇢ small enough, ⌘S 2 ND(m0).
  42. Recovery of Positive Measures Theorem: let and [de Castro et

    al. 2011] ! m0 is recovered when there is no noise. ⌘S(t) = 1 ⇢ QN i=1 sin(⇡(t xi))2 -1 1 -1 1 -1 1 -1 1 ⌘S ⌘S ⌘S ⌘S [Morgenshtern, Cand` es, 2015] discrete `1 robustness. [Demanet, Nguyen, 2015] discrete `0 robustness. Input measure: m0 = ma,x where a 2 RN + . ! behavior as 8 i, xi ! 0 ? '(x) = (ei`x)fc `= fc for N 6 fc and ⇢ small enough, ⌘S 2 ND(m0).
  43. Recovery of Positive Measures Theorem: let and [de Castro et

    al. 2011] ! m0 is recovered when there is no noise. ! noise robustness of support recovery ? ⌘S(t) = 1 ⇢ QN i=1 sin(⇡(t xi))2 -1 1 -1 1 -1 1 -1 1 ⌘S ⌘S ⌘S ⌘S [Morgenshtern, Cand` es, 2015] discrete `1 robustness. [Demanet, Nguyen, 2015] discrete `0 robustness. Input measure: m0 = ma,x where a 2 RN + . ! behavior as 8 i, xi ! 0 ? '(x) = (ei`x)fc `= fc for N 6 fc and ⇢ small enough, ⌘S 2 ND(m0).
  44. Asymptotic of Vanishing Certificate 1 ⌘V Vanishing Derivative pre-certificate: ⌘V

    def. = argmin ⌘= ⇤p ||p|| m0 = ma, x where ! 0 s.t. 8 i, ⇢ ⌘( xi) = 1, ⌘0( xi) = 0. Valid only in 1-D, i.e. T = R or T = R/Z.
  45. Asymptotic of Vanishing Certificate 1 ⌘V 1 1 1 ⌘V

    ⌘V ⌘W Vanishing Derivative pre-certificate: ⌘V def. = argmin ⌘= ⇤p ||p|| m0 = ma, x where ! 0 s.t. 8 i, ⇢ ⌘( xi) = 1, ⌘0( xi) = 0. Valid only in 1-D, i.e. T = R or T = R/Z.
  46. Asymptotic of Vanishing Certificate 1 ⌘V 1 1 1 ⌘V

    ⌘V ⌘W s.t. ⇢ ⌘(0) = 1, ⌘0(0) = . . . = ⌘(2N 1)(0) = 0. Asymptotic pre-certificate: ⌘W def. = argmin ⌘= ⇤p ||p|| Vanishing Derivative pre-certificate: ⌘V def. = argmin ⌘= ⇤p ||p|| ! 0 m0 = ma, x where ! 0 s.t. 8 i, ⇢ ⌘( xi) = 1, ⌘0( xi) = 0. Valid only in 1-D, i.e. T = R or T = R/Z.
  47. Asymptotic Robustness Theorem: the solution of P (y) for y

    = (m0) + w is for w , w 2N 1 , 2N 1 = O(1) PN i=1 a? i x? i ||w || N = 2 ||w || N = 1 If ⌘W 2 NDN , letting m0 = ma, x, then where ||(x, a) (x ? , a ?)|| = O ✓ ||w|| + 2N 1 ◆ [Denoyelle, D., P. 2015]
  48. Asymptotic Robustness Theorem: the solution of P (y) for y

    = (m0) + w is x? i 0 ↵ < 2N 1 Noise: w = w0. for w , w 2N 1 , 2N 1 = O(1) PN i=1 a? i x? i Regularization: = 0 ↵ 0 max ↵ = 2N 1 x? i x0 ||w || N = 2 ||w || N = 1 If ⌘W 2 NDN , letting m0 = ma, x, then where ||(x, a) (x ? , a ?)|| = O ✓ ||w|| + 2N 1 ◆ y = ma, x + w [Denoyelle, D., P. 2015]
  49. Gaussian convolution: Proposition: ⌘W (x) = e x2 4 2

    N 1 X k=0 (x/2 )2k k! In particular, ⌘W is non-degenerate. 1 N = 1 ⌘W 1 N = 3 ⌘W '(x) = e |x ·|2 2 2 ! Gaussian deconvolution <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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is support-stable. <latexit 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<latexit 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Deconvolution and Laplace
  50. Gaussian convolution: Proposition: ⌘W (x) = e x2 4 2

    N 1 X k=0 (x/2 )2k k! In particular, ⌘W is non-degenerate. 1 N = 1 ⌘W 1 N = 3 ⌘W '(x) = e |x ·|2 2 2 ! Gaussian deconvolution <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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is support-stable. <latexit 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<latexit 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Deconvolution and Laplace Laplace transform: '(x) = e x· Non-translation-invariant operator ¯ x x1 x2 ! ⌘W depends on ¯ x! Proposition: In particular, ⌘W is non-degenerate. ⌘W (x) = 1 ✓ x ¯ x x + ¯ x ◆2N N = 1 N = 3 ⌘W ⌘W ¯ x = 2 ¯ x = 20
  51. Higher Dimension Gaussian convolution: m0 = ma, x where !

    0 The limiting certificate in multi-dimensions The limit now depends on the spikes configuration! t = 1 t = 0.5 t = 0.2 t = 0.01 15 / 31 ⌘W <latexit 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⌘V <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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= 0 <latexit 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on the directions x! <latexit 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⌘W depends <latexit 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<latexit 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  52. Higher Dimension Gaussian convolution: m0 = ma, x where !

    0 The limiting certificate in multi-dimensions The limit now depends on the spikes configuration! t = 1 t = 0.5 t = 0.2 t = 0.01 15 / 31 ⌘W <latexit 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⌘V <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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= 0 <latexit 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on the directions x! <latexit 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⌘W depends <latexit 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<latexit 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<latexit sha1_base64="7bfhbu2Pa9Qt1lkGoWmyrirvzIc=">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</latexit> Proposition: <latexit 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<latexit 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⌘W def. = argmin ⌘= ⇤p ||p|| s.t. ⇢ ⌘(0) = 1, P1(@)⌘(0) = . . . = P(d+1)N 1 (@)⌘(0) = 0. <latexit 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</latexit> <latexit 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</latexit> 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</latexit> 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</latexit> ⌘V !0 ! ⌘W where <latexit 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<latexit 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De Boor basis <latexit 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<latexit 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<latexit 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Carl R. de Boor [Poon, P. 2018] <latexit 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<latexit 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  53. Higher Dimension Gaussian convolution: m0 = ma, x where !

    0 The limiting certificate in multi-dimensions The limit now depends on the spikes configuration! t = 1 t = 0.5 t = 0.2 t = 0.01 15 / 31 ⌘W <latexit 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⌘V <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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= 0 <latexit 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on the directions x! <latexit 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⌘W depends <latexit 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<latexit 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<latexit sha1_base64="7bfhbu2Pa9Qt1lkGoWmyrirvzIc=">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</latexit> Proposition: <latexit 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<latexit 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⌘W def. = argmin ⌘= ⇤p ||p|| s.t. ⇢ ⌘(0) = 1, P1(@)⌘(0) = . . . = P(d+1)N 1 (@)⌘(0) = 0. <latexit 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</latexit> <latexit 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49dmpCeq3nheuqJxdPNtQrL81h4z9OXwA13p/E9AqxXnn2/UHJDS4PWufCmgk3KIav8PAf4Ev49hr8u/XJNHes8PIt6ThR9+nVRCj3xenLrypXmSglHdQ/UrnqojqCN76h3jep4XnqqZJd+quwbr7iWTMpcfUD4kP8akY5iaa5bNH1U+xDf5+86ewnyYK0vWbVkzx0Uc5+TrbAPlPzVslZRbSuem5FTCMU65Di+/oNS8RwSz7vzOBh3a/JJlfzNcJOcfDITUpTNLmle67eh7bDeXe26TZyk37BX9Lpqall7uQR5pC2juRljTisvRbShYXIW66raWl3aKN6MV9fCbLuVlaq6tvZ9sbT/73ZWJf+yQ/5v08I4sqrny25WWNgfbOd/23bexUPdfCcLuqqv+oMt/V+1pav6IduO/LswulTDfKOX67zJpHDLXEvfMid3b4XGgXkOsuc9tWKmhpUqK/5mhSyPz8civj2uV+Nnoqu6dzLHjMH5xIHctWnuxO3rFSiO39o0zyD2PKd4bkqtg+/gbSk5BfuhkruRq5GjEupOLjouYlrEbUoaLSI/yMaT+ftIcV80ziDYpzdk9jDRJ7Hwdjq01CSHXMvoiX2F6Bap8e3dMns3KmiYT3eYG4STXPnyeWWvLUePSSaRkVF2Tl9m0iWecvO+KJFKqOf5893ITcqJ1G9n9G3ZjAQdWrGs0lVCVu3Tpjnxsql473mP6phPoMwU7qqdOThWc3p3Dk3SCc9j2X5U5g2KGjb5y56XKbIfHZJniKFo8pcpVs0iFcuN7WNTe+ekEuXWpo2t5lhVpwsrfg5xjqEhWnTTCdUWUtqIohSqpTwllzdOdD2iLF1q5W2ySt63N6Uxw0T7Wd4xhNoN36O+qMz3MucT+a7/AekRd0O0dJmreBRzVp/z4PMVU0KZd3MGT9rYfXVCO3cTPRuFc72XnlL2abZtqrp6HUc0u68eqIfq10CrC7m/9pZiqvMINintx/LvpzjMbvOzdwiHb0faUs+yOxqr9hHKO49Snul73vx0dwNUy3R3M7p+yiF5fRL7KT9QT2jcUEcbLPVJxI7ih3T3mo8y3jR3QrkapRvyq+neBxkelOpeUkM7kmSvFN8ELdLYtwFepxOEfO8C2uG0lnQtSovjYnJehdOBnrl1nd8y34V0+VTrsqNvAfLlvtS7IkN7IS+pFJzzlZJ9ueMg/YmS85gyxr6ks9kyojbxQphK/syQOUnJZbBpyrg9RPcyk89n05yH7zybq+J9Yi5Lj+EpdyKxPvJUixqOobaXo8d2y3uzixLHU2faB1nJDe08xZWM25zKFVefB2pk0V0B5GqkPYl/K9pADFc+V2R/XrF09ZZu66rSXVg++c2gId2eaKy1yrbw9kVXa6jjNQzPRUmGeWZnRauNo3jPSdPW18Jra+HdpNUyS3/ko3AEfQ3feHwU0XdxTtNP4V2i+dYQHmsIlS06c/HSQeM4SOVeRgUtoKsp5ekcR5Sex6b5c0fuM9n+06qt7K5VvP30hGrgVu3aPFDyWw3mE37vo+enyDu3eUwrbb5jnfLd0BESxzfmlBSe/7hNaS/gYSs90WfBE/WrkiQ2ekYjj1fqpkoyn+KvA74TSnpSO23FooTjq9Uo+7Kt63GkNRlb8ufdzvI+VXE3HCJCMPZtA3kK8dLuw6jOTY+lj6d0pOmEWv5+xi8/ohoRt8cB9Oeq6q7Zz9XC2RuKxfB8kfwCkP3rPCNCmNMszDufH38xCFP6YMkDin9nqt6vNyEF3smDqB166+pdP+wbhjR6xghN7oQwvz/Ep8+xpeGslf9sGp46HyrXLHHxt0h43ofP5qCUfM7sJZRpSGfdP9Ie6zjCOwyIQvF2HJtDi+aYeGcBjp0fqH8KUn1NumkWTmaF5H+coZJCGUJnINBb81mDED+7NC+hXdrYMle5/WkD7GcP/F69U1MdrVeJIcInVs9y8htfbG6BONHeqhVRs1X3Wcf8XsNB1t7Ld79U3Ri9ApjQqPWg5EdsGjEUZqp4fzS3j1mwVHKLYREZ3nXYUcU72LleOhHW2aNxXa/QFvIS3KOyr+tRLNb0rSyi81F/QfdPs38+D/ZbL6z8+HYCvYj7zGv++5jZA4kGOGovWqkP/VbHjis0vnHFCn7shUbXiz/CcZX8hkcxFnT1Ve7YzW8XAyXng4s3OftOOslq0U7GwT2P2FHmFzaZKp/lbVBaSrVtrLBDn/XZ/5P3lzeKv/Rafnm+ubaxvrbx6Z3l397TvwL7I/XX6u9AFxvqQ/Vb9Qhs8ghk+q9rf3PtxrVfXPznm79687dvfs5Zf3BNY36mcv+9ufk/HCZnFQ==</latexit> 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</latexit> 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</latexit> ⌘V !0 ! ⌘W where <latexit 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<latexit 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De Boor basis <latexit 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<latexit 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<latexit 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Carl R. de Boor [Poon, P. 2018] <latexit 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<latexit 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<latexit 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Non-degeneracy of ⌘W is necessary for stable recovery. <latexit 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<latexit 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<latexit 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necessary and su cient for N = 2 spikes. <latexit 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sha1_base64="Ns1PRMzXH2i39aVpmBqR0z/Yb3w=">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</latexit> Proposition: <latexit 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<latexit 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  54. Gaussian Mixtures + s N = 2 N = 2

    Clustering about point (m0, s0) = (0, 2) For N = 2, ⌘W,Z is a valid certificate if |m1 So, one cannot expect to super-resolve variation in the means is too large with Gaussian mixture For x = (m, s) 2 X = R ⇥ R⇤ + , '(x) = 1 s e (· m)2 2s2 2 H = L 2( N = 2 N = 2 N = 2 Clustering about point (m0, s0) = (0, 2). y-axis = me For N = 2, ⌘W,Z is a valid certificate if |m1 m2| 6 |s1 s2| . So, one cannot expect to super-resolve with BLASSO a mixt variation in the means is too large with respect to variations sian mixture = (m, s) 2 X = R ⇥ R⇤ + , '(x) = 1 s e (· m)2 2s2 2 H = L 2(R). N = 2 N = 2 N = 2 N = 2 ring about point (m0, s0) = (0, 2). y-axis = mean, x-axis = standard = 2, ⌘W,Z is a valid certificate if |m1 m2| 6 |s1 s2| . e cannot expect to super-resolve with BLASSO a mixture of 2 Gaussians w on in the means is too large with respect to variations in the standard devia ⇤ + , '(x) = 1 s e (· m)2 2s2 2 H = L 2(R). N = 2 N = 2 N = 2 µ <latexit 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<latexit sha1_base64="kx/AuIG3LNOfw+zA1cXFgxUqzds=">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</latexit> '(x) = 1 e (· µ)2 2 2 <latexit 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<latexit 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<latexit 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Gaussian mixtures: <latexit 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X = R ⇥ R+, <latexit 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<latexit 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<latexit sha1_base64="7UmOgJFSl1i5m9Bfx+DJMCn5ggs=">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</latexit> ⌘W is non-degenerate if m 6 . <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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  55. Gaussian Mixtures + s N = 2 N = 2

    Clustering about point (m0, s0) = (0, 2) For N = 2, ⌘W,Z is a valid certificate if |m1 So, one cannot expect to super-resolve variation in the means is too large with Gaussian mixture For x = (m, s) 2 X = R ⇥ R⇤ + , '(x) = 1 s e (· m)2 2s2 2 H = L 2( N = 2 N = 2 N = 2 Clustering about point (m0, s0) = (0, 2). y-axis = me For N = 2, ⌘W,Z is a valid certificate if |m1 m2| 6 |s1 s2| . So, one cannot expect to super-resolve with BLASSO a mixt variation in the means is too large with respect to variations sian mixture = (m, s) 2 X = R ⇥ R⇤ + , '(x) = 1 s e (· m)2 2s2 2 H = L 2(R). N = 2 N = 2 N = 2 N = 2 ring about point (m0, s0) = (0, 2). y-axis = mean, x-axis = standard = 2, ⌘W,Z is a valid certificate if |m1 m2| 6 |s1 s2| . e cannot expect to super-resolve with BLASSO a mixture of 2 Gaussians w on in the means is too large with respect to variations in the standard devia ⇤ + , '(x) = 1 s e (· m)2 2s2 2 H = L 2(R). N = 2 N = 2 N = 2 µ <latexit 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<latexit sha1_base64="kx/AuIG3LNOfw+zA1cXFgxUqzds=">AABBw3ictVzbchu5EYU3t7Vz8yaPeZlE65Q35TiS46qkaitVK0uyrDVty0tK9u7SdvEyomkNOfQMKV+4+oS8Jp+T78gfJE/5hfQFGGBIzDRGcYyShAFxuhs9QKO7Abo/S8b5fHPzn5c++t73f/DDH318+cqPf/LTn/386ie/OM7TRTaIjwZpkmZP+708TsbT+Gg+nifx01kW9yb9JH7SP93Bz5+cxVk+Tqed+btZ/GzSG03HJ+NBbw5N7W4+fnF1Y/PmJv2L1itburKh9L/D9JPoSHXVUKVqoBZqomI1VXOoJ6qncijfqi21qWbQ9kwtoS2D2pg+j9W5ugLYBfSKoUcPWk/h9wievtWtU3hGmjmhB8AlgZ8MkJG6BpgU+mVQR24Rfb4gythaRXtJNFG2d/C3r2lNoHWuXkKrhDM9Q3E4lrk6UX+mMYxhTDNqwdENNJUFaQUlj5xRzYHCDNqwPoTPM6gPCGn0HBEmp7Gjbnv0+b+oJ7bi80D3Xah/k5TXoESqrUefFhR66ozoR/Q2F/AZy5MA5xFQiPUYsfaGdD2h0U+h/xLaH0I5p5rRSR/KklrPa5E7UHzIHRG5D8WH3BeRLSg+ZEtEHkLxIQ81ErEZ6dyPb0Px4dsi58dQfMjHIvIrKD7kVyLyGIoPeSwiv4HiQ34jIu9C8SHvisj7UHzI+yKyA8WH7IjIIyg+5JGI3IPiQ+5pZPVKzaCkRGcsrMptqJd5oKVIoGVblO8OWUcf9k7Amh5UYOVVvQt//djdAJ3GFdi9gHl3UoGVZ94+2Eg/VrZF92g38WHvidgDmAF+7IGI/VK9qsB+GbDSTiuw8lprQT8/Vra+D+DJj30gYh9CzY+V96hH0OLHPgrYMWYV2EMR+1i9rsCGWP2sAivb/TbYFT9W3qc60N+PDbGmiwqsbE+PwYPxY+Xd6gm0+rFPROxT9bYC+1TEfg3W3Y/9OmCHfV+BNXvsFdpBRuSPxLBi66j1ilWJtRlQ6wn8k2JvScg37kO7hBkVmBFhJiJiv0DsByJaBaIVLFde2NGc/F2ZS7tAtAMR/WJvwtpc7D8s+mMtCUDsFojdFUSdR4rv2ozljLwL0yIh58XOhbWQMaWF/cZarOdDveU1iEclBM/tlzTzb1C0hBEUaqqO2stij2dkRM91iDcUvZlRGh4ybl5YBRf1VkT1Pai+iHrnQb0TUQsPaiGizjyoMxFlV76L6wbMAKt/fBdLeuIZwD5ydYnAK9iGXecerNEI5s8heIFfUcsj+Num2FsqdZJhNI/7JGY5npUscQa1pdqAdhsV7lJ8ndAKi0Ey7vlIx/j4hLmNpV5zbIXPi508KjIm4XTGJM+ooIPeYkTrqRmd+9RyTt4d15rh7xXr3tSa4fdI4+fkxXOtGX6upZ9fQPaOxnYugG3Dappp7dt6Uxqcf2Eapn6Fdl20uPhWJ3rOIL23Dekf6DdzcIH3skM11o+tN6ORO+PLS+NrQsPqOXf03IwKek/s9Zpa1HgkUx332npTGVLaRadaDvvU9M1gn6F+M6bejMYheFw7FHMvnXrT2TsrRmPrzWgcK857npMnb+rNaIzomfVh681oYLalp+N8W29q2VEDHDvbelOrPqUsMOaAeM5zi/WKMvKTFpramPyD+myN6/Ov72OYs3lexAj1lKxvW02nX+xl9RIZfyEGqzZvKAf6FwvHByvTWKpbYnzFMsxL+/s6HbvHo+ZboMUIVj+fAUg58wQkNDkJtN4JUNwSo67yyAzulojDWXKygurq1rnoLVq+nDUqt72gVikus6O1euySvc5p7s3IJ2yRZiU9tCrfcBVFSUOtkoZkek10916v17L2N0XcbAUxK2bagE6E+CStPk71ab3t6PiaPuWZQ+EzHzt/Mdt8oq0Nxjwp2SKUpY6n28/kkdw23FdvKJvj5s8ieqNor87IaozpRCoXo1CTLWZvfEnPlvYRnckhD6YxgPcYaSozxadmmEXHfHpEFtW1txJv1JfJ0HE9J6tr7HE9euSgRx508xhnB3aMh1DrQMxwBE+dgCjnSqGrlDSeqd8Xp6MpvcH6iD4pWUhDg+1NXLKQdVH2yxKVN4DG2cBRejiNVToG312jJEf9Pnls7Fq2/Nfo5Nacb/dojlfP5upMzJC43iKuEa0aPtXlp1UOLMHS+8kt8l/rR4n8mnBEGypxfe5wZr1M6cQ/pgh2Rp5xQqtNWh3l3m5+avUTw+lQmbNzPM1OyUJGZP8i2J9SmpMR/bh3B8wJOluEhGxkiN0ZF96Nz9cZi3PM+nFjxbca7HyLyZYtiL+h666unOYiRwy8D5yvzG2jkxb5gjFxzbR1t2u7fvdBpL0n4c4SpmjnynXi/xn9Nj9mnmyszQjUML6BXNs63/tIKWZBHfVol6+3QaavK+WnhQzPtdR2/7MyfVqSbJciLpQHd+shcB7QM/PCWZKR3PlaH95H67K5SHm2okcc7QlF8Wz3R3oHRrlv0C65QWuuS7NkBLNgXkQRpq+URV7lW8+rTD2Mdv5/oW51XdYaUoyUzeCyhqT8fkzRmitlArOa5+8prSa/1rOVXvV8pjQXJ85a/g5afw2/jdzmOYxOv2QV7tAcYAr2yWqEW6K1HmG87pR4mZlpaNlny8/OSdPLbblIfM3WzcbYZ42pHNKseauzFqZ+ERqvHBqvAnXYobNGq0XTbizRCzG26OjTylB+Tbh1GlBeiJRlj8ygxgFSurFUGNWhSFWO8Q3qvUhrU6TVg9Xqnga4az4E6V/rq6v7u2J3j9Rd8m0G5IFx/DKkVTomn8u01kdqTAE539b21V39XWpB7n2yoEiZ73HiiuFTpwGV80LS3+qdLSU7by2Cubf0RvcxNrZL9T+uISe0JnJalwZxm3rEWn5XjmjFIt10fI6IMv898qnY76iPmd3e9p1EJX/Cxpu8qiwvjhSmpH8p83awFr0eOPFrRDHhQnvXfaDV/A0jBcaYTILfs8zpDeEuxycJ7NH2yX6u2yk+xZs6Et0kqZfqLwE2hqNeO9fduWVGbMb2O+iJWrdv3ddD5pcEc5T4XeREr0e72kT7qMuV54vR6uldrvxcp4fFCl+rjwX1cSMLG+WVMV31eTAXlqgZF8aEcGk2iibyN5O8icx8OhVK2fQ2lMuZBrYxLyleku6BIsLn3V33enOfCePor9HrE9alxi0SJczGpTo/4FpazEpdXtuHuPVy7W6UODtR1U5hqLu7hbXfbCFjsn6JknI23NuVvVuKUuQsDFMYKL7RWxUfujQ/h4K/I+WLDg3HkNxhG/zbbbWj9j7AbYjXus4ZzYha0BYMV2Lvnh5nuUe9jl471F36IRzCeYxB15L0Y9pJm8rOlGXJXerh9N+QFchULEpvezYfg8tFHsk6pybjGZNlk0czVua7OE3HYjiEjKTMJZwPn2tIozhR5jtNzcZgqMsjKHNowsPcYwh757Z3c14up3p9rXMJ5cG7gDlxMTg8+auOVWy/EAuVOW/kw3NA63BSQ93sFv/rOAwfy6k5r1BuOX3X7FXAW+d+sc7Ioj/cfM1YbiGzuZpjOM+0GJ31lvz82O+LGr2p1BnNh6eP/qidA4bXUnEeVJaO8e4ssvKGUsFzAZ8MqfqP+scl+dsIrwsaVXI0oWTOKaqpmR4yNfONS9/ozGchMlk6VTKVqdk4ok03YnfUgboLPzuFB9j0dih/l5L/Itb//dkhtJ6Q9TBZdM4cdKktpuyHPUUb0rO9P1slMd7l5bu9HWjBs/AWteI934fUH+/6dkpjq/4GCa/1BypVw1JEsnq6Z9dVH0ZQPnnjHJD5nm9Ed+k5i8U3zyYBZ4vm/tSqREv6RL5Z0K/E9x0pBzRXZ/qsHk8O8IZ9r8gPReoP1NbTdh73XInzYSXnwxXOOWmnzOGt81n93awqLjsOl2GROzvT/VKKs+15Xn1udLeSC99Br8ePavAjR8o2af+UIuFM1WfzFjU0F1om94R1qkwmkvWAcWaveN/1ke1ZDa+zgPHfr0TfdyTdB1n6lP+O6IQtI3qJ1s0eSc83HeszqfdqpNXfo3xxdWNr9f8yWK8c37q5tXlz6/HtjS/u6P/n4GP1K/UbdR3W+J/UF0DtUB0Bh5H6q/qb+vv23vbpdrY9564fXdKYX6rSv+3z/wKlvqFw</latexit> '(x) = 1 e (· µ)2 2 2 <latexit 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<latexit 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<latexit 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Gaussian mixtures: <latexit 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X = R ⇥ R+, <latexit 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<latexit 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<latexit sha1_base64="7UmOgJFSl1i5m9Bfx+DJMCn5ggs=">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</latexit> ⌘W is non-degenerate if m 6 . <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Neuro-imaging Let X = x 2 R2 ; kxk 6 1 . To model MEG/EEG, '(x) = u 7! kx uk 2 2 H = L 2(@X) N = 1 N = 2 N = 2 N = 3 Neuro-imaging: <latexit 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<latexit 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X = x 2 R2 ; ||x|| 6 1 , <latexit 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<latexit 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<latexit 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'(x) = (||x z|| 2)||z||=1 <latexit 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<latexit 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  56. Conclusion varying the azimuth φ during the ex modeled by

    the following expression: gðθÞ = Z2π 0 Z∞ 0 Z∞ −∞ Iðz; α; φÞρ  θ Ω= A B C D Fig. 2. Experimental validation of the mult of the system designed to create a slope of of the maximum intensity projection of imag TIRF illumination. (Scale bar: 5 μm.) The evolu illumination angle θ of two selected beads a sponding fitting theoretical model (continu depth (respectively 10 and 89 nm). (D) Depth o fitting the theoretical TIRF model (in red) and estimated by fitting a Gaussian model in the [Boulanger et al. 2014] Super-resolution should be o↵-the-grid! More accurate theory: <latexit 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<latexit 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More e cient algorithms. <latexit 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sha1_base64="jIpTXtgscy3iBRd86UPdpj3CN4c=">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</latexit> – “`2” error meaningless. <latexit 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<latexit 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– Track the support. <latexit 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<latexit 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<latexit 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– Fisher-Rao + Optimal Transport. <latexit 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<latexit 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<latexit 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<latexit 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  57. Conclusion Open problem: other regularizations (e.g. piecewise constant) ? see

    [Chambolle, Duval, Peyr´ e, Poon 2016] for TV denoising. varying the azimuth φ during the ex modeled by the following expression: gðθÞ = Z2π 0 Z∞ 0 Z∞ −∞ Iðz; α; φÞρ  θ Ω= A B C D Fig. 2. Experimental validation of the mult of the system designed to create a slope of of the maximum intensity projection of imag TIRF illumination. (Scale bar: 5 μm.) The evolu illumination angle θ of two selected beads a sponding fitting theoretical model (continu depth (respectively 10 and 89 nm). (D) Depth o fitting the theoretical TIRF model (in red) and estimated by fitting a Gaussian model in the [Boulanger et al. 2014] Super-resolution should be o↵-the-grid! More accurate theory: <latexit 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<latexit 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More e cient algorithms. <latexit 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sha1_base64="jIpTXtgscy3iBRd86UPdpj3CN4c=">AABB3HictVzNchu5EYY3P7t2/rzJLblMonXKm3IUSdmqpGorVStLsqw1bcsmJXt3abv4M6LGHnLoGVL+4aoqh9xSueYRck3eIM+RN0hOeYV0N4ABhsRMA4rXKEkYEF93owdodDdA96dpUsw2Nv516b1vffs7333/g8tXvvf9H/zwR1c//PFxkc3zQXw0yNIsf9zvFXGaTOKjWTJL48fTPO6N+2n8qP9iBz9/dBbnRZJNOrM30/jJuDeaJCfJoDeDpmdXf3o3y+MoPoGWJJ7Mol46yvJkdjou1p9dXdtY36B/0WplU1XWhPp3mH0YHYmuGIpMDMRcjEUsJmIG9VT0RAHlK7EpNsQU2p6IBbTlUEvo81iciyuAnUOvGHr0oPUF/B7B01eqdQLPSLMg9AC4pPCTAzIS1wCTQb8c6sgtos/nRBlb62gviCbK9gb+9hWtMbTOxCm0cjjd0xeHY5mJE/F7GkMCY5pSC45uoKjMSSsoeWSNagYUptCG9SF8nkN9QEit54gwBY0dddujz/9NPbEVnweq71z8h6S8BiUSbTX6rKTQE2dEP6K3OYfPpDwpcB4BhViNEWuvSNdjGv0E+i+g/R6Uc6ppnfShLKj1vBG5A8WF3GGR+1BcyH0W2YLiQrZY5CEUF/JQIRGbk87d+DYUF77Ncn4AxYV8wCIfQnEhH7LIYygu5DGL/BKKC/kli7wFxYW8xSLvQHEh77DIDhQXssMij6C4kEcscg+KC7mnkPUrNYeSEZ2EWZXbUK/yQEuRQss2K99Nso4u7E2PNT2owfKrehf+urG7HjqNa7B7HvPupAbLz7x9sJFuLG+LbtNu4sLeZrEHMAPc2AMW+7l4XoP93GOlvajB8mutBf3cWN763oUnN/Yui70HNTeW36PuQ4sbe99jx5jWYA9Z7APxsgbrY/XzGixv99tgV9xYfp/qQH831seazmuwvD09Bg/GjeV3q0fQ6sY+YrGPxesa7GMW+wVYdzf2C48d9m0NVu+xV2gHGZE/EsOKbaLWK1cl1qZArcfwT8u9JSXfuA/tHGZUYkaEGbOI/RKx74lolYiWt1xFaUcL8nd5Lu0S0fZE9Mu9CWsztv+w7I+11AOxWyJ2lxBNHim+az2WM/IudAuHnJU7F9Z8xpSV9htrsZoPzZZXI+5XEHJun9LMv0HREkZQqKkmaqflHi+RET03IV5R9KZHqXnwuFlpFWzUaxbVd6D6LOqNA/WGRc0dqDmLOnOgzliUWfk2rusxA4z+8V0s6EnOAOkj15cIvIJt2HVuwxqNYP4cghf4kFruw982xd5caZIMo3ncJzHL8aRiiXOoLcQatJuocJfi65RWWAySyZ73VYyPT5jbWKg1J63webmTR2XGxJ9OQvKMSjroLUa0nsLo3KGWc/LuZC0Mf7tc97oWht8jjZ+TFy9rYfiZkn52Adk7Ctu5ALYNq2mqtG/qoTRk/kXS0PUrtOuixcW3OlZzBum9DqR/oN7MwQXeyw7VpH5MPYxGYY2vqIwvhIbRc2HpOYwKek/S69W1KHgkExX3mnqoDBntohMlh3kKfTPYZ6jejK6H0TgEj2uHYu6FVQ+dvdNyNKYeRuNYyLznOXnyuh5GY0TPUh+mHkYDsy09FeebeqhlRw3I2NnUQ636hLLAmAOSc162GK8oJz9prqgl5B80Z2tsn391H8OczdMyRmimZHzbejr9ci9rlkj7CzFYtVmgHOhfzC0frEpjIbbY+ErKMKvs76t0zB6Pmm+BFiNY/fIMgMuZpyChzkmg9U6B4iYbdVVHpnFbLA5nyckSqqtaZ6y3aPjKrFG17Rm1cnGZGa3RY5fsdUFzb0o+YYs0y+mhVfuG6yhyGmpVNMTTC9HdW7Veq9rfYHHTJcS0nGkDOhGSJ2nNcapL621Lx9fUKc8MijzzMfMXs80nytpgzJORLUJZmnja/XQeyW7DffWGMDlu+VlEbxTt1RlZjYROpAo2CtXZYumNL+jZ0D6iMznkIWkM4D1GispUyFMzzKJjPj0ii2rbW4436ktn6GS9IKur7XEzemShRw50eIyzAzvGPah1IGY4gqeOR5RzpdRVRhrPxa/L09GM3mBzRJ9WLKSmIe1NXLGQTVH2aYXKK0DjbJBRuj+NZToa312hxEf9LnlM7Fq1/Nfo5Fafb/dojtfP5vpMzJC4bhHXiFaNPNWVT8scpAQL5ydb5L82jxL5hXBEG8pxfWpxlnqZ0Il/TBHslDzjlFYbtzqqve381PInmtOh0GfneJqdkYWMyP5FsD9lNCcj+rHvDugTdGkRUrKRPnYnKb0bl6+TsHPM+HGJkLcazHyLyZbNib+ma6+uguaijBjkPnC+NLe1TlrkC8bENVfW3azt5t0HkeaehD1LJEUzV64T/4/pt/7R82RtZUaghvENFMrWud5HRjEL6qhHu3yzDdJ9bSk/KmV4qqQ2+5+R6aOKZLsUcaE8uFsPgfOAniUvnCU5yV2s9JH7aFM2FylPl/SIoz2hKF7a/ZHagVHuG7RLrtGa69IsGcEsmJVRhO7LZZGX+TbzqlL3o118I9SNrqtaQ4qRMBlcqSEuvx9TtGZLmcKslvP3Ba0mt9bzpV7NfCY0F8fWWv4aWn8Ov7Xc+tmPTr9iFW7SHJAUzJPRiGyJVnr48bpZ4aVnpqZlng0/Myd1L7vlIvG1tG4mxj4LpnJIs+a1ylro+kVoPLdoPPfUYYfOGo0Wdbu2RM/Y2KKjTit9+YVw6wRQnrOUeY9MoxIPKe1Yyo/qkKXKx/ga9ZaltcHS6sFqtU8D7DXvg3Sv9eXV/XW5u0fiFvk2A/LAZPwypFWakM+lW5sjNUkBOX+i7Ku9+rvUgtz7ZEGRsrzHiStGnjoNqJyXkv5S7WwZ2XljEfS9pVeqj7axXar/dgU5pjVR0LrUiE+oR6zkt+WIlizSuuVzRJT575FPJf2O5pjZ7m3eSVTxJ0y8KVeV4SUjhQnpn8u8HaxErwdW/BpRTDhX3nUfaIW/YaQgMTqT4PYsC3pDuMvJkwTp0fbJfq7aKXmKN7EkWiepF+IPHjZGRr1mrttzS49Yj+1X0BO1bt66qwfPL/XmyPG7yIlej3a1sfJRF0vPF6PVU7tc9blJD/MlvkYfc+pjRxYmyqtiuuJTby5SojAuEuPDJWwUIfKHSR4iszyd8qWse2vK1UyDtDGnFC9x90AR4fLurju9uY+ZcfRX6PUJa1OTLRwlzMZlKj9gW1rMSl1e2Ydk6+XG3Si1dqK6nUJTt3cLY7+lhYzJ+qWCy9nI3rbs3UqUwmdhJIWBkDd66+JDm+anUPB3JFzRoebokztsg3+7LXbE3ju4DfFS1WVGM6IWtAXDpdi7p8ZZ7dGso5cWdZu+Dwd/HgnompM+oZ00VHZJmZfcpu5P/xVZgVzErPSmZ/gYbC78SFY5hYwnIcvGjyYR+rs4oWPRHHxGUuXiz0eea3CjOBH6O01hY9DU+RFUOYTw0PcY/N656R3Oy+bUrK9VLr485C6gT1w0Dk/+6mMV08/HQuXWG3n3HNA6nDRQ17vF/zsOzcdwCufly62g75o993jrsl+sMrLoD4evGcPNZzbXc/TnmZWjM96Sm5/0+6KgN5VZo3n39NEfNXNA81oImQflpZN4exYZeX2p4LmAS4ZM/Ff88xL/bYSXJY06OUIo6XOKemq6B09Nf+PSNTr9mY9Mhk6dTFVqJo5o043YHXEgbsHPTukBht4Old+llH8R6/7+7BBaT8h66Cy6zBx0qS2m7Ic5RRvSs7k/Wycx3uWVd3s70IJn4S1qxXu+96g/3vXtVMZW/w0SudbvikwMKxHJ8umeWVd9GEH15E3mgPT3fCO6Sy+zWPLm2djjbFHfn1qWaEGf8DcL+rX4viXlgObqVJ3V48kB3rDvlfmhSPyG2nrKzuOey3E+rOV8uMS5IO1UOby2Pmu+m1XHZcfiMixzZ2eqX0ZxtjnPa86N7tZykXfQm/GjBvzIkrJN2n9BkXAumrN58waacyWTfcI6EToTKfWAcWavfN/Nke1ZA68zj/HfqUXfsSTdB1n6lP+O6IQtJ3qp0s0eSS9vOjZnUm83SKu+R/ns6trm8v9lsFo53lrf3FjffLC19tlN9f8cfCB+Jn4hrsMa/534DKgdiiPg8EfxN/F38Y/tp9t/2v7z9l9k1/cuKcxPROXf9l//B1wXqzU=</latexit> – “`2” error meaningless. <latexit 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<latexit 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– Track the support. <latexit 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<latexit 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<latexit 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– Fisher-Rao + Optimal Transport. <latexit 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<latexit 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<latexit 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<latexit 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