Off-the-Grid Sparse Estimation

E34ded36efe4b7abb12510d4e525fee8?s=47 Gabriel Peyré
November 08, 2019

Off-the-Grid Sparse Estimation

Talk @Toulouse "Workshop on Computational aspects of Geometry"

E34ded36efe4b7abb12510d4e525fee8?s=128

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=">AABBzXictVzbchu5EYU3t7Vz8yaPeZlE65Q35XUkZauSylaqVpZkWWuuRZuU7N2l7eJlRNMecmgOKV9o5TWfkNfkS/Id+YPkKb+QvgADDImZxiiOUZIwIE53owdodDdA96bJKJtvbv7z0gff+e73vv+DDy9f+eGPfvyTn1796GcnWbqY9ePjfpqks0e9bhYno0l8PB/Nk/jRdBZ3x70kfth7sYufPzyLZ9konbTnb6bx43F3OBmdjvrdOTQ9bgCsO4vG6SBO/vj06sbmzU36F61XtnRlQ+l/zfSj6Fh11EClqq8WaqxiNVFzqCeqqzIo36ottamm0PZYLaFtBrURfR6rc3UFsAvoFUOPLrS+gN9DePpWt07gGWlmhO4DlwR+ZoCM1DXApNBvBnXkFtHnC6KMrWW0l0QTZXsDf3ua1hha5+oZtEo40zMUh2OZq1P1BxrDCMY0pRYcXV9TWZBWUPLIGdUcKEyhDesD+HwG9T4hjZ4jwmQ0dtRtlz7/F/XEVnzu674L9W+S8hqUSLX06NOcQledEf2I3uYCPmN5EuA8BAqxHiPWXpGuxzT6CfRfQvs9KOdUMzrpQVlS63klcheKD7krIg+g+JAHIrIBxYdsiMgmFB+yqZGInZHO/fgWFB++JXK+D8WHvC8iH0DxIR+IyBMoPuSJiPwGig/5jYi8DcWHvC0i70LxIe+KyDYUH7ItIo+h+JDHInIfig+5r5HlK3UGJSU6I2FV7kC9yAMtRQItO6J8t8g6+rC3AtZ0vwQrr+o9+OvH7gXoNC7B7gfMu9MSrDzzDsBG+rGyLbpDu4kPe0fEHsIM8GMPReyX6nkJ9suAlfaiBCuvtQb082Nl6/sVPPmxX4nYe1DzY+U96gha/NijgB1jWoJtitj76mUJNsTqz0qwst1vgV3xY+V9qg39/dgQa7oowcr29AQ8GD9W3q0eQqsf+1DEPlKvS7CPROzXYN392K8Ddti3JVizx16hHWRI/kgMK7aKWjdflVibArWuwD/J95aEfOMetEuYYY4ZEmYsIg5yxEEgopEjGsFyZbkdzcjflbm0ckQrENHL9yaszcX+g7w/1pIAxF6O2FtBVHmk+K7NWM7IuzAtEnKe71xYCxlTmttvrMV6PlRbXoM4KiB4bj+jmX+DoiWMoFBTVdSe5Xs8IyN6rkK8oujNjNLwkHHz3Cq4qNciqudB9UTUGw/qjYhaeFALEXXmQZ2JKLvyXVwnYAZY/eO7WNITzwD2kctLBF7BDuw6d2CNRjB/muAFPqCWI/jbothbKlWSYTSP+yRmOR4XLPEMaku1Ae02Ktyj+DqhFRaDZNzzSMf4+IS5jaVec2yFz/OdPMozJuF0RiTPMKeD3mJE66kenbvUck7eHdfq4e/k697U6uH3SePn5MVzrR5+rqWfX0D2tsa2L4BtwWqaau3bel0anH9hGqZ+hXZdtLj4Vsd6ziC91zXpH+o3c3iB97JLNdaPrdejkTnjywrjq0PD6jlz9FyPCnpP7PWaWlR7JBMd99p6XRlS2kUnWg77VPfNYJ+BfjOmXo9GEzyuXYq5l0697uyd5qOx9Xo0ThTnPc/Jkzf1ejSG9Mz6sPV6NDDb0tVxvq3XteyoAY6dbb2uVZ9QFhhzQDznucV6RTPykxaa2oj8g+psjevzr+9jmLN5kscI1ZSsb1tOp5fvZdUSGX8hBqs2rykH+hcLxwcr0liqbTG+Yhnmhf19nY7d41HzDdBiBKufzwCknHkCEpqcBFrvBChuiVFXcWQGty3icJacrqA6unUueouWL2eNim1PqVWKy+xorR47ZK8zmntT8gkbpFlJD43SN1xGUdJQo6AhmV4d3b3V67Wo/U0RN11BTPOZ1qcTIT5Jq45TfVpvOTq+pk955lD4zMfOX8w2n2prgzFPSrYIZani6fYzeSS3DffVG8rmuPmziN4o2qszshojOpHKxCjUZIvZG1/Ss6V9TGdyyINp9OE9RprKVPGpGWbRMZ8ekUV17a3EG/VlMnRcz8jqGntcjR466KEHXT/G2YUd4x7U2hAzHMNTOyDKuZLrKiWNz9Sn+eloSm+wOqJPChbS0GB7ExcsZFWU/axA5RWgcTZwlB5OY5WOwXfWKMlRv08eG7sWLf81Ork159tdmuPls7k8EzMgrtvENaJVw6e6/LTKgSVYej/ZJv+1epTIrw5HtKES1ycOZ9bLhE78Y4pgp+QZJ7TapNVR7O3mp1Y/MZyaypyd42l2ShYyIvsXwf6U0pyM6Me9O2BO0NkiJGQjQ+zOKPdufL7OSJxj1o8bKb7VYOdbTLZsQfwNXXd1ZTQXOWLgfeB8ZW4bnTTIF4yJ60xbd7u2q3cfRNp7Eu4sYYp2rlwn/p/Qb/Nj5snG2oxADeMbyLSt872PlGIW1FGXdvlqG2T6ulJ+nMvwREtt9z8r08cFyfYo4kJ5cLceAOc+PTMvnCUzkjtb68P7aFU2FylPV/SIoz2lKJ7t/lDvwCj3DdolN2jNdWiWDGEWzPMowvSVssirfKt5FamH0c7+L9StrotaQ4qRshlc1pCU348pWnOlTGBW8/x9QavJr/XZSq9qPhOai2NnLb+D1l/CbyO3eQ6j0ytYhVs0B5iCfbIa4ZZorUcYr1sFXmZmGlr22fKzc9L0clsuEl+zdbMx9lltKk2aNa911sLUL0LjuUPjeaAO23TWaLVo2o0leirGFm19WhnKrw63dg3KC5Gy7JEZ1ChASjeWCqM6EKnKMb5BvRVpbYq0urBa3dMAd82HIP1rfXV1v8t390jdJt+mTx4Yxy8DWqUj8rlMa3WkxhSQ82favrqrv0MtyL1HFhQp8z1OXDF86tSncp5L+mu9s6Vk561FMPeWXuk+xsZ2qP67NeSY1kRG69IgPqMesZbflSNasUg3HZ8josx/l3wq9juqY2a3t30nUcGfsPEmryrLiyOFCelfyrwdrkWvh078GlFMuNDedQ9o1X/DSIExJpPg9ywzekO4y/FJAnu0PbKf63aKT/EmjkQ3Seql+lOAjeGo1851d26ZEZux/QZ6otbtW/f1kPklwRwlfhc50evSrjbWPupy5flitLp6lys+V+lhscLX6mNBfdzIwkZ5RUxHfR7MhSWqx4UxIVzqjaKO/PUkryMzn06FUja9DeVipoFtzDOKl6R7oIjweXfXvd7cJ8I4emv0eoR1qXGLRAmzcanOD7iWFrNSl9f2IW69XLkbJc5OVLZTGOrubmHtN1vImKxfoqScDfd2Ze8UohQ5C8MU+opv9JbFhy7Nz6Hg70j5okPDMSR32AL/dkftqv33cBvipa5zRjOiFrQFg5XYu6vHWexRraOXDnWXfgiHcB4j0LUk/Yh20rqyM2VZcpd6OP1XZAVmKhaltz3rj8HlIo9knVOd8YzIssmjGSnzXZy6YzEcQkZS5BLOh881pFGcKvOdpnpjMNTlERQ51OFh7jGEvXPbuz4vl1O1vta5hPLgXcCcuBgcnvyVxyq2X4iFmjlv5P1zQOtwWkHd7Bb/6zgMH8upPq9Qbhl91+x5wFvnfrHOyKI/XH/NWG4hs7mcYzjPNB+d9Zb8/Njvi2q9qdQZzfunj/6onQOG11JxHlSWjvHuLLLyhlLBcwGfDKn6j/rHJfnbCC9zGmVy1KFkzinKqZkeMjXzjUvf6MxnITJZOmUyFanZOKJFN2J31aG6DT+7uQdY93Yof5eS/yLW//3ZAbSekvUwWXTOHHSoLabshz1FG9CzvT9bJjHe5eW7vW1owbPwBrXiPd971B/v+rYLYyv/Bgmv9a9UqgaFiGT1dM+uqx6MoHjyxjkg8z3fiO7ScxaLb56NA84Wzf2pVYmW9Il8s6BXiu85UvZprk71WT2eHOAN+26eH4rUb6mtq+087rkS52Yp5+YK54y0U+Tw2vms+m5WGZddh8sgz52d6X4pxdn2PK86N7pXyoXvoFfjhxX4oSNli7T/giLhmarO5i0qaC60TO4J60SZTCTrAePMbv6+qyPbswpeZwHjv1uKvutIegCy9Cj/HdEJ24zoJVo3+yQ933SszqTeqZBWf4/y6dWNrdX/y2C9crJ9c2vz5tb97Y0vbun/5+BD9Qv1K3Ud1vjv1RdAramOyd7+Vf1N/X3naGex827nz9z1g0sa83NV+Lfzl/8CF2GlYA==</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=">AABBx3ictVzdchPJFW42fwv5Y5PL3EziJcWmCLEJVUnV1latsY3xIkAg2Xh3BZR+xkIw0giNZAxaX+QRcps8TJ4jb5Bc5RVyfrqne6SeOT0Ooct2T6u/c06f6T59zukWvWkyyuabm/+89NH3vv+DH/7o48tXfvyTn/7s51c/+cVRli5m/fiwnybp7LjXzeJkNIkP56N5Eh9PZ3F33Evip73XO/j509N4lo3SSXv+bho/G3eHk9HJqN+dQ9PTzvHZF50nT15c3di8uUn/ovXKlq5sKP2vmX4SHaqOGqhU9dVCjVWsJmoO9UR1VQblW7WlNtUU2p6pJbTNoDaiz2N1rq4AdgG9YujRhdbX8HsIT9/q1gk8I82M0H3gksDPDJCRugaYFPrNoI7cIvp8QZSxtYz2kmiibO/gb0/TGkPrXL2EVglneobicCxzdaL+TGMYwZim1IKj62sqC9IKSh45o5oDhSm0YX0An8+g3iek0XNEmIzGjrrt0uf/op7Yis993Xeh/k1SXoMSqZYefZpT6KpToh/R21zAZyxPApyHQCHWY8TaW9L1mEY/gf5LaH8I5ZxqRic9KEtqPa9E7kDxIXdE5D4UH3JfRDag+JANEdmE4kM2NRKxM9K5H9+C4sO3RM6PofiQj0XkEyg+5BMReQTFhzwSkd9A8SG/EZF3ofiQd0XkfSg+5H0R2YbiQ7ZF5CEUH/JQRO5B8SH3NLJ8pc6gpERnJKzKbagXeaClSKBlW5TvDllHH/ZOwJrul2DlVb0Lf/3Y3QCdxiXYvYB5d1KClWfePthIP1a2RfdoN/Fh74nYA5gBfuyBiP1KvSrBfhWw0l6XYOW11oB+fqxsfR/Akx/7QMQ+hJofK+9Rj6DFj30UsGNMS7BNEftYvSnBhlj9WQlWtvstsCt+rLxPtaG/HxtiTRclWNmeHoEH48fKu9VTaPVjn4rYY3VWgj0WsV+Ddfdjvw7YYd+XYM0ee4V2kCH5IzGs2Cpq3XxVYm0K1LoC/yTfWxLyjXvQLmGGOWZImLGI2M8R+4GIRo5oBMuV5XY0I39X5tLKEa1ARC/fm7A2F/sP8v5YSwIQuzlidwVR5ZHiuzZjOSXvwrRIyHm+c2EtZExpbr+xFuv5UG15DeJRAcFz+yXN/BsULWEEhZqqovYy3+MZGdFzFeItRW9mlIaHjJvnVsFFnYmongfVE1HvPKh3ImrhQS1E1KkHdSqi7Mp3cZ2AGWD1j+9iSU88A9hHLi8ReAXbsOvcgzUawfxpghf4hFoewd8Wxd5SqZIMo3ncJzHL8axgiWdQW6oNaLdR4S7F1wmtsBgk456PdIyPT5jbWOo1x1b4PN/JozxjEk5nRPIMczroLUa0nurRuU8t5+Tdca0e/l6+7k2tHn6PNH5OXjzX6uHnWvr5BWRva2z7AtgWrKap1r6t16XB+RemYepXaNdFi4tvdaznDNI7q0n/QL+Zgwu8lx2qsX5svR6NzBlfVhhfHRpWz5mj53pU0Htir9fUotojmei419brypDSLjrRctinum8G+wz0mzH1ejSa4HHtUMy9dOp1Z+80H42t16NxpDjveU6evKnXozGkZ9aHrdejgdmWro7zbb2uZUcNcOxs63Wt+oSywJgD4jnPLdYrmpGftNDURuQfVGdrXJ9/fR/DnM3zPEaopmR923I6vXwvq5bI+AsxWLV5TTnQv1g4PliRxlLdEuMrlmFe2N/X6dg9HjXfAC1GsPr5DEDKmScgoclJoPVOgOKWGHUVR2Zwt0QczpKTFVRHt85Fb9Hy5axRse0FtUpxmR2t1WOH7HVGc29KPmGDNCvpoVH6hssoShpqFDQk06uju/d6vRa1vynipiuIaT7T+nQixCdp1XGqT+stR8fX9CnPHAqf+dj5i9nmE21tMOZJyRahLFU83X4mj+S24b56Q9kcN38W0RtFe3VKVmNEJ1KZGIWabDF740t6trQP6UwOeTCNPrzHSFOZKj41wyw65tMjsqiuvZV4o75Mho7rGVldY4+r0UMHPfSg68c4O7BjPIRaG2KGQ3hqB0Q5V3JdpaTxmfp9fjqa0husjuiTgoU0NNjexAULWRVlvyxQeQtonA0cpYfTWKVj8J01SnLU75PHxq5Fy3+NTm7N+XaX5nj5bC7PxAyI6y3iGtGq4VNdflrlwBIsvZ/cIv+1epTIrw5HtKES1+cOZ9bLhE78Y4pgp+QZJ7TapNVR7O3mp1Y/MZyaypyd42l2ShYyIvsXwf6U0pyM6Me9O2BO0NkiJGQjQ+zOKPdufL7OSJxj1o8bKb7VYOdbTLZsQfwNXXd1ZTQXOWLgfeB8ZW4bnTTIF4yJ60xbd7u2q3cfRNp7Eu4sYYp2rlwn/p/Rb/Nj5snG2oxADeMbyLSt872PlGIW1FGXdvlqG2T6ulJ+msvwXEtt9z8r06cFyXYp4kJ5cLceAOc+PTMvnCUzkjtb68P7aFU2FylPV/SIoz2hKJ7t/lDvwCj3DdolN2jNdWiWDGEWzPMowvSVssirfKt5FamH0c7+L9StrotaQ4qRshlc1pCU348pWnOlTGBW8/x9TavJr/XZSq9qPhOai2NnLX8Hrb+G30Zu8xxGp1ewCndoDjAF+2Q1wi3RWo8wXncKvMzMNLTss+Vn56Tp5bZcJL5m62Zj7NPaVJo0a8501sLUL0LjlUPjVaAO23TWaLVo2o0leiHGFm19WhnKrw63dg3KC5Gy7JEZ1ChASjeWCqM6EKnKMb5BvRdpbYq0urBa3dMAd82HIP1rfXV1f5fv7pG6S75Nnzwwjl8GtEpH5HOZ1upIjSkg59vavrqrv0MtyL1HFhQp8z1OXDF86tSncp5L+lu9s6Vk561FMPeW3uo+xsZ2qP7HNeSY1kRG69IgblOPWMvvyhGtWKSbjs8RUea/Sz4V+x3VMbPb276TqOBP2HiTV5XlxZHChPQvZd4O1qLXAyd+jSgmXGjvuge06r9hpMAYk0nwe5YZvSHc5fgkgT3aHtnPdTvFp3gTR6KbJPVSfRFgYzjqtXPdnVtmxGZsv4OeqHX71n09ZH5JMEeJ30VO9Lq0q421j7pceb4Yra7e5YrPVXpYrPC1+lhQHzeysFFeEdNRnwdzYYnqcWFMCJd6o6gjfz3J68jMp1OhlE1vQ7mYaWAb85LiJekeKCJ83t11rzf3mTCO3hq9HmFdatwiUcJsXKrzA66lxazU5bV9iFsvV+5GibMTle0Uhrq7W1j7zRYyJuuXKClnw71d2TuFKEXOwjCFvuIbvWXxoUvzcyj4O1K+6NBwDMkdtsC/3VY7au8D3IZ4o+uc0YyoBW3BYCX27upxFntU6+iNQ92lH8IhnMcIdC1JP6KdtK7sTFmW3KUeTv8tWYGZikXpbc/6Y3C5yCNZ51RnPCOybPJoRsp8F6fuWAyHkJEUuYTz4XMNaRQnynynqd4YDHV5BEUOdXiYewxh79z2rs/L5VStr3UuoTx4FzAnLgaHJ3/lsYrtF2KhZs4b+fAc0DqcVFA3u8X/Og7Dx3KqzyuUW0bfNXsV8Na5X6wzsugP118zllvIbC7nGM4zzUdnvSU/P/b7olpvKnVG8+Hpoz9q54DhtVScB5WlY7w7i6y8oVTwXMAnQ6r+o/5xSf42wpucRpkcdSiZc4pyaqaHTM1849I3OvNZiEyWTplMRWo2jmjRjdgddaDuws9O7gHWvR3K36Xkv4j1f392AK0nZD1MFp0zBx1qiyn7YU/RBvRs78+WSYx3eflubxta8Cy8Qa14z/ch9ce7vu3C2Mq/QcJr/YFK1aAQkaye7tl11YMRFE/eOAdkvucb0V16zmLxzbNxwNmiuT+1KtGSPpFvFvRK8T1Hyj7N1ak+q8eTA7xh383zQ5H6A7V1tZ3HPVfi3Czl3FzhnJF2ihzOnM+q72aVcdlxuAzy3Nmp7pdSnG3P86pzo7ulXPgOejV+WIEfOlK2SPuvKRKeqeps3qKC5kLL5J6wTpTJRLIeMM7s5u+7OrI9reB1GjD++6Xo+46k+yBLj/LfEZ2wzYheonWzR9LzTcfqTOq9Cmn19yhfXN3YWv2/DNYrR7dubm3e3Hp8e+PLO/r/OfhY/Ur9Rl2HNf4n9SVQa6pDsmx/VX9Tf98+2E63T7fPuOtHlzTml6rwb/sv/wUjoaLJ</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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