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Random Polynomials in some Learning Problems

Random Polynomials in some Learning Problems

UniRandom presentation on Random Polynomials in some Learning Problems.
Based on "Testing Gaussian Process with Applications to Super-Resolution" (with J.-M. Azaïs & S. Mourareau), Applied and Computational Harmonic Analysis.

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Yohann De Castro

September 10, 2019
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  1. Yohann DE CASTRO (École Centrale Lyon & Institut Camille Jordan)

    Random Polynomials in some Learning Problems Unirandom with Jean-Marc AZAÏS (IMT) and Stéphane MOURAREAU
  2. Yohann DE CASTRO 3 higher than one fluorophore per mm

    . We present a novel method by which fluorescence micros- copy may be performed to obtain an image with greatly enhanced ability to resolve large numbers of fluorescent activated (fluorescent) state m light-controlled activation ra 2. For irreversible photoactiv quantum yield must be fini FIGUR localiza area co (here, neously for rea spatial i a secon diode l Within vation b blue cir circles) time, th (red Xs (black then activated, localized, and bleached until a sufficient number of molecules have been analyzed to construct an image. (G the 405-nm activation laser (X405), which is reflected by a dichroic (DM1) to make it collinear with the Ar1 readout lase inverted fluorescence microscope is used to focus the lasers, which are reflected upward by a second dichroic mirror (D objective lens (OBJ). The sample, supported by a coverslip (CS), emits fluorescence which is collected by the objective, tra and focused by the tube lens (TL) to form an image on a camera (CCD). Biophy S. Hess, T. Girirajan, M. Mason, Ultra-High Resolution Imaging by Fluorescence Photoactivation Localization Microscopy, Biophysical Journal (2004). Sparse Deconvolution
  3. Yohann DE CASTRO 4 Y. Li, S. Osher, R. Tsai,

    Heat Source Identification based on L1 Constrained Minimization, Inverse Problems and Imaging (2014). 210 Yingying Li, Stanley Osher and Richard Tsai (a) Heat source u0 (b) Au0 = f (c) Sparse Deconvolution
  4. Yohann DE CASTRO 5 H. Pan, T. Blu, M. Vetterli,

    Towards Generalized FRI Sampling With an Application to Source Resolution in Radioastronomy, IEEE trans. on Signal Processing (2017). nuous-time al problem, occurrence y’s method. resolution sparse sig- ling theory estimation But not all y a concrete develop the ally sum of identifying rm samples m samples id solution n such that Fig. 1. Schematic diagram of a radio interferometer. The cros of the received signals at different antennas are related to the Fou of the sky image (see Table I) at certain non-uniform frequencies Sparse Deconvolution
  5. Yohann DE CASTRO 6 New Optimization Paradigm x <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit>

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  6. Yohann DE CASTRO 6 New Optimization Paradigm x <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit>

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  7. Yohann DE CASTRO 6 New Optimization Paradigm x <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit>

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  8. Yohann DE CASTRO 6 New Optimization Paradigm x <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">AAACxHicjVHLSsNAFD2Nr/quunQTLIKrkoigy6IgLluwD6hFknRaQ6dJmJmIpegPuNVvE/9A/8I74xTUIjohyZlz7zkz994w47FUnvdacObmFxaXissrq2vrG5ulre2mTHMRsUaU8lS0w0AyHiesoWLFWTsTLBiFnLXC4ZmOt26ZkHGaXKpxxrqjYJDE/TgKFFH1u+tS2at4ZrmzwLegDLtqaekFV+ghRYQcIzAkUIQ5Akh6OvDhISOuiwlxglBs4gz3WCFtTlmMMgJih/Qd0K5j2YT22lMadUSncHoFKV3skyalPEFYn+aaeG6cNfub98R46ruN6R9arxGxCjfE/qWbZv5Xp2tR6OPE1BBTTZlhdHWRdclNV/TN3S9VKXLIiNO4R3FBODLKaZ9do5Gmdt3bwMTfTKZm9T6yuTne9S1pwP7Pcc6C5mHF9yp+/ahcPbWjLmIXezigeR6jigvU0DDej3jCs3PucEc6+WeqU7CaHXxbzsMHaxyPfQ==</latexit>

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  9. Yohann DE CASTRO 7 Mathematical Modeling x <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit

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  10. Yohann DE CASTRO 7 Mathematical Modeling x <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit

    sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> (x) <latexit sha1_base64="D5K3UcQJmwIeX6hFpf2q0mjsMiE=">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</latexit> y = (x) + " <latexit sha1_base64="c/WH112Xy41jtfZThibZGM00peE=">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</latexit> Target Discrete Measure ⌫0 <latexit sha1_base64="PnqprQ8RIm0yaZQQq2XHdGCzcQw=">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</latexit> F(⌫0) <latexit sha1_base64="7GjpsO/qiPyYOcWpiI1zoNPhXHQ=">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</latexit> F(⌫0) + " <latexit sha1_base64="+50kg6D9NApF8/b3gjiqs7T3Bo4=">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</latexit> Fourier Coefficients n <latexit sha1_base64="XAAt5PrCMHtkuCnwbCw9cJJzyVw=">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</latexit> " 2 Cn <latexit sha1_base64="ALyEtZSNr6D0hOhpRoIw5J5D8hY=">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</latexit>
  11. Yohann DE CASTRO 7 Mathematical Modeling x <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit

    sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> (x) <latexit sha1_base64="D5K3UcQJmwIeX6hFpf2q0mjsMiE=">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</latexit> y = (x) + " <latexit sha1_base64="c/WH112Xy41jtfZThibZGM00peE=">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</latexit> Target Discrete Measure ⌫0 <latexit sha1_base64="PnqprQ8RIm0yaZQQq2XHdGCzcQw=">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</latexit> F(⌫0) <latexit sha1_base64="7GjpsO/qiPyYOcWpiI1zoNPhXHQ=">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</latexit> F(⌫0) + " <latexit sha1_base64="+50kg6D9NApF8/b3gjiqs7T3Bo4=">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</latexit> Fourier Coefficients n <latexit sha1_base64="XAAt5PrCMHtkuCnwbCw9cJJzyVw=">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</latexit> " 2 Cn <latexit sha1_base64="ALyEtZSNr6D0hOhpRoIw5J5D8hY=">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</latexit> Low Rank Tensor M0 <latexit sha1_base64="dhcZ9YaTUtKM7yqMe81DvwN5zak=">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</latexit> M0 + " <latexit sha1_base64="i3f4wQlav2G2P51b87+AgP+RizI=">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</latexit> " 2 Rn⇥n <latexit sha1_base64="0YIBCF7Haf+NJw91b3PPuw3xwa0=">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</latexit> M0 <latexit sha1_base64="dhcZ9YaTUtKM7yqMe81DvwN5zak=">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</latexit>
  12. Yohann DE CASTRO 9 Mathematical Modeling Z0 = F?F(⌫0) Z

    = Z0 + F?(") Z0 = Z ⇢(· t)d⌫0(t) <latexit sha1_base64="SSubjj2IQPsOoGwHT6yUotj7Iww=">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</latexit> ⇢ the nth Dirichlet kernel <latexit sha1_base64="tT7z3L7Isi3arYovekcXpfbLV0w=">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</latexit> with F?(") nth centered Gaussian <latexit sha1_base64="8IATvSJFowxH5/eA8vt3C3+X8hU=">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</latexit> random polynomial <latexit sha1_base64="vpXzH52ys/TfK/b0d7asZZNujII=">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</latexit>
  13. Yohann DE CASTRO 11 Statistical Learning Observation X(t, ✓) =

    X0(t, ✓) + Re(e ı✓(F?")(t)) max ✓ X(·, ✓)
  14. Yohann DE CASTRO 11 Statistical Learning Observation X(t, ✓) =

    X0(t, ✓) + Re(e ı✓(F?")(t)) max ✓ X(·, ✓)
  15. Yohann DE CASTRO 12 Statistical Learning 1 b ⌫( )

    = 0 = 1 = 10.66 for X( )(t, ✓) = X(t, ✓) ( 1 )⇢(t t1, ✓ ✓1) b ⌫( ) 2 arg min ⌫ n1 2 ky F(⌫)k2 2 + k⌫kT V o Z( )(t) = Z(t) ( 1 ) F?F(b ⌫( ))
  16. Yohann DE CASTRO 13 Statistical Learning for X( )(t, ✓)

    = X(t, ✓) ( 1 )⇢(t t1, ✓ ✓1) b ⌫( ) 2 arg min ⌫ n1 2 ky F(⌫)k2 2 + k⌫kT V o Z( )(t) = Z(t) ( 1 ) F?F(b ⌫( )) = 10.36 b ⌫( )
  17. Yohann DE CASTRO 14 Statistical Learning for X( )(t, ✓)

    = X(t, ✓) ( 1 )⇢(t t1, ✓ ✓1) b ⌫( ) 2 arg min ⌫ n1 2 ky F(⌫)k2 2 + k⌫kT V o Z( )(t) = Z(t) ( 1 ) F?F(b ⌫( )) b ⌫( ) = 10.06
  18. Yohann DE CASTRO 15 Statistical Learning for X( )(t, ✓)

    = X(t, ✓) ( 1 )⇢(t t1, ✓ ✓1) b ⌫( ) 2 arg min ⌫ n1 2 ky F(⌫)k2 2 + k⌫kT V o Z( )(t) = Z(t) ( 1 ) F?F(b ⌫( )) b ⌫( ) = 9.76
  19. Yohann DE CASTRO 16 Statistical Learning = 2 = 9.49

    2 for X( )(t, ✓) = X(t, ✓) ( 1 )⇢(t t1, ✓ ✓1) b ⌫( ) 2 arg min ⌫ n1 2 ky F(⌫)k2 2 + k⌫kT V o Z( )(t) = Z(t) ( 1 ) F?F(b ⌫( )) b ⌫( )
  20. Yohann DE CASTRO 17 Statistical Learning 2 1 b ⌫(

    ) 2 arg min ⌫ n1 2 ky F(⌫)k2 2 + k⌫kT V o b ⌫( ) LARS « knots » of the LASSO
  21. Yohann DE CASTRO 17 Statistical Learning 2 1 b ⌫(

    ) 2 arg min ⌫ n1 2 ky F(⌫)k2 2 + k⌫kT V o b ⌫( ) LARS « knots » of the LASSO
  22. Yohann DE CASTRO 20 One Spike Detection ⌫0 = a1

    t1 <latexit sha1_base64="cWCKPaqr1nM1vHGBVw2a+ALpExw=">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</latexit> One Spike Model
  23. Yohann DE CASTRO 20 One Spike Detection ⌫0 = a1

    t1 <latexit sha1_base64="cWCKPaqr1nM1vHGBVw2a+ALpExw=">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</latexit> One Spike Model Observation F?(") = f X k= f ("1 k + ı"2 k ) exp(ık·) <latexit sha1_base64="ClM/YIiCgDfxnntjbNxq6y0YkwU=">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</latexit> "1,2 k ⇠i.i.d. N(0, 1/n) <latexit sha1_base64="hyWaDhlz5rPnUUAFunNbftUtKW0=">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</latexit> n = 2f + 1 <latexit sha1_base64="4uBHfQpnAqBt8NL1hi567YwrNyM=">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</latexit> with and Z = a1⇢(· t1) + F?(") <latexit sha1_base64="btitX/Jx5wTSQDBIe8AvSQMIWQo=">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</latexit>
  24. Yohann DE CASTRO 20 One Spike Detection ⌫0 = a1

    t1 <latexit sha1_base64="cWCKPaqr1nM1vHGBVw2a+ALpExw=">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</latexit> One Spike Model Maximum Likelihood b t1 2 arg max t |Z(t)| <latexit sha1_base64="E2k9p/eQK6bYYshP2bpW4Beu944=">AAAC5nicjVHLLgRBFD3a+z1Y2jQTCZtJ95CwFDaWJAahZVLdU2Yq+pXqao/MzNrOTmz9gC2fIv6Av3CrtMQjQnW6+9S595yqe6+fhiJTjvPcY/X29Q8MDg2PjI6NT0yWpqb3siSXAa8FSZjIA59lPBQxrymhQn6QSs4iP+T7/ummju+fcZmJJN5Vlyk/jlgzFiciYIqoemnOOxcN3mKqrepu1xOxx2TTi9hFXdmdw0W11KmXyk7FMcv+CdwClFGs7aT0BA8NJAiQIwJHDEU4BENGzxFcOEiJO0abOElImDhHFyOkzSmLUwYj9pS+TdodFWxMe+2ZGXVAp4T0SlLaWCBNQnmSsD7NNvHcOGv2N++28dR3u6S/X3hFxCq0iP1L95H5X52uReEEa6YGQTWlhtHVBYVLbrqib25/qkqRQ0qcxg2KS8KBUX702TaazNSue8tM/MVkalbvgyI3x6u+JQ3Y/T7On2CvWnGXK9WdlfL6RjHqIcxiHos0z1WsYwvbqJH3Fe7xgEerZV1bN9bte6rVU2hm8GVZd2+dTp0t</latexit> Observation F?(") = f X k= f ("1 k + ı"2 k ) exp(ık·) <latexit sha1_base64="ClM/YIiCgDfxnntjbNxq6y0YkwU=">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</latexit> "1,2 k ⇠i.i.d. N(0, 1/n) <latexit sha1_base64="hyWaDhlz5rPnUUAFunNbftUtKW0=">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</latexit> n = 2f + 1 <latexit sha1_base64="4uBHfQpnAqBt8NL1hi567YwrNyM=">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</latexit> with and Z = a1⇢(· t1) + F?(") <latexit sha1_base64="btitX/Jx5wTSQDBIe8AvSQMIWQo=">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</latexit>
  25. Yohann DE CASTRO 21 Statistical Learning thin grids Grid-less approach

    Grid approach SRice SST from the Hessian and from ( 1, 2, R) R LARS from grid points LARS from the process X(·) testing procedures X(·) (X(tk))p k=1 ( 1,p, 2,p) p
  26. Yohann DE CASTRO 22 The Independent Part of the Hessian

    e ⇤ = ⇢00(0) where and ⇢ covariance function of X X00(z) = e ⇤X(z) | {z } E ⇥ X00(z) (X(z),X0(z)) ⇤ +R(z)
  27. Yohann DE CASTRO 22 The Independent Part of the Hessian

    e ⇤ = ⇢00(0) where and ⇢ covariance function of X X00(z) = e ⇤X(z) | {z } E ⇥ X00(z) (X(z),X0(z)) ⇤ +R(z) 10 3 LOOKING AT JOINT DENSITIES: THE RICE METHOD where Leb(R) denotes the Lebesgue measure on R. As a consequence we can prove the following proposition. Proposition 5. The joint law L(( 1 , 2 , R(b z))) of ( 1 , 2 , R(b z)) satisfies for all (`1 , `2 , r) 2 R2 ⇥S, dL(( 1 , 2 , R(b z))) dLeb(R) ⌦ µ (`1 , `2 , r) = (cst) det( e ⇤`1 + r)1{0<`2<`1} 1 ( 1`1) , where Leb(R) ⌦ µ is defined by (15) and S denotes the set of symmetric matrices. Proof. Observe that the density at point `1 of X(0) with respect to the Lebesgue measure is 1 ( 1`1) and recall (15). Now, for any Borel set B of R2 ⇥ S, note that E h det( e ⇤X(0) + R(0))1{(0< 0 2 <X(0)}1{(X(0), 0 2 ,R(0))2B} i = Z B det( e ⇤`1 + r)1{0<`2<`1} 1 ( 1`1)d`1 µ (d(`2 , r)) µ where law of ⇣ max y nE[X(y)|(X(z), X0(z))] 1 ⇢(y z) o , R(z) ⌘ and fixed ˆ z s.t. X(ˆ z) = 1 where z
  28. Yohann DE CASTRO 23 The Independent Part of the Hessian

    can show that under the alternative H1 P(( 1 , 2 , R(z)) 2 B) = (cst) Z T E(| det( e ⇤X(z) + R(z))|10< z 2 <X(z)1(X(z), z 2 ,R(z))2B )dz = (cst) Z B⇥T det( e ⇤l1 + r)10<l2<l1 1(l1 µ(z)) µ ,z(d(l2 , r)) ✓q µ0(z)T e ⇤ 1µ0(z) ◆ dl1dz where µ(·) = E(X(·)) and µ0 denotes the gradient of µ. We can now state our result when the variance is known. Theorem 6. Set 8r 2 S +, 8` > 0, Gr(`) := Z +1 ` det( e ⇤u + r) (u 1)du , where e ⇤ is the Hessian of the correlation function ⇢ of X at the origin. Under Assumptions (Anorm ) and (Adegen ), the test statistic SRice := GR(b z) ( 1) GR(b z) ( 2) ⇠ U([0, 1]) under the null H0 . Proof. Using Proposition 5, we know that the density of 1 at `1 and conditional to ( 2 , R(b z)) = (`2 , r) is equal to (cst) det( e ⇤`1 + r) ( 1`1)1{0<`2<`1} , It is well known that, if a random variable Z has for cumulative density function F then F(Z) follows an uniform distribution on [0, 1]. This implies that, conditionally to ( 2 , R(b z)) = (`2 , r),
  29. Yohann DE CASTRO 24 Better Detection thin grids Grid-less approach

    Grid approach Inference from grid points Inference from the process X(·) p
  30. Yohann DE CASTRO 24 Better Detection thin grids Grid-less approach

    Grid approach Inference from grid points Inference from the process X(·) p Grids Off-the-grid
  31. Yohann DE CASTRO 26 One Spike Detection in Spiked Tensor

    PCA One Spike Model Observation Likelihood y = M0 + " and "i1,...,id ⇠iid N(0, 1/n) <latexit sha1_base64="S+0B2RBdLNdG7Uv14O2axi0PZQU=">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</latexit> hy, x⌦di = ht, xid + X i1,...,id "i1,...,id xi1 . . . xid <latexit sha1_base64="qY6RyxtRVckXKRMSWakQisYqGtk=">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</latexit> M0 = t⌦d and t 2 Rn , ktk2 = 1 <latexit sha1_base64="XDAlHOOpLLeI5ZquJFMyytIscJQ=">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</latexit> *sphere taken from Dmitry Belyaev’s webpage
  32. Yohann DE CASTRO 26 One Spike Detection in Spiked Tensor

    PCA One Spike Model Observation Likelihood y = M0 + " and "i1,...,id ⇠iid N(0, 1/n) <latexit sha1_base64="S+0B2RBdLNdG7Uv14O2axi0PZQU=">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</latexit> hy, x⌦di = ht, xid + X i1,...,id "i1,...,id xi1 . . . xid <latexit sha1_base64="qY6RyxtRVckXKRMSWakQisYqGtk=">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</latexit> Information Bound: b t !n!1 t , = O(1) <latexit sha1_base64="pd0fVavLc/28p6n8UpPjwQTwY4g=">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</latexit> Computational Bound: = O(nd 2 4 ) <latexit sha1_base64="yvupiLgnxu6AK22N+kH7vbjrIdU=">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</latexit> exponentially large number of critical points close to the maximum M0 = t⌦d and t 2 Rn , ktk2 = 1 <latexit sha1_base64="XDAlHOOpLLeI5ZquJFMyytIscJQ=">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</latexit> *sphere taken from Dmitry Belyaev’s webpage
  33. Yohann DE CASTRO 26 One Spike Detection in Spiked Tensor

    PCA One Spike Model Observation Likelihood y = M0 + " and "i1,...,id ⇠iid N(0, 1/n) <latexit sha1_base64="S+0B2RBdLNdG7Uv14O2axi0PZQU=">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</latexit> hy, x⌦di = ht, xid + X i1,...,id "i1,...,id xi1 . . . xid <latexit sha1_base64="qY6RyxtRVckXKRMSWakQisYqGtk=">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</latexit> Information Bound: b t !n!1 t , = O(1) <latexit sha1_base64="pd0fVavLc/28p6n8UpPjwQTwY4g=">AAADIXicjVFNb9QwEJ2mBUr56EKPXCxWSOWySgpSuSBVcOFQiSJ120p1tXIc765VJ47sCVW06q/hn3DrDfVSVf0DcIRf0LFJVaBC4CjJ+M17bzyevDbaY5qezyXzC7du31m8u3Tv/oOHy71Hj3e8bZxUQ2mNdXu58MroSg1Ro1F7tVOizI3azQ/fhvzuR+W8ttU2trU6KMWk0mMtBRI06nF+pAs1FciQ8UL72ojWY2sUN7aaOD2ZonDOHo1mFUfLdTXG9jhwN9UYr9OMG6pZiNe8FDiVwrD3q9nzUa+fDtK42M0g64I+dGvL9s6AQwEWJDRQgoIKkGIDAjw9+5BBCjVhBzAjzFGkY17BMSyRtiGWIoYg9JC+E9rtd2hF++Dpo1pSFUOvIyWDZ6SxxHMUh2os5pvoHNC/ec+iZzhbS/+88yoJRZgS+i/dFfN/daEXhDG8ij1o6qmOSOhOdi5NvJVwcvZLV0gONWEhLijvKJZReXXPLGp87D3crYj5r5EZ0LCXHbeBb+GUNODsz3HeDHbWBtmLwdqHl/2NN92oF+EJPIVVmuc6bMA72IIheX+mat/hR/IpOUm+JKc/qclcp1mB31ZycQlJ4rZF</latexit> Computational Bound: = O(nd 2 4 ) <latexit sha1_base64="yvupiLgnxu6AK22N+kH7vbjrIdU=">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</latexit> exponentially large number of critical points close to the maximum M0 = t⌦d and t 2 Rn , ktk2 = 1 <latexit sha1_base64="XDAlHOOpLLeI5ZquJFMyytIscJQ=">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</latexit> *sphere taken from Dmitry Belyaev’s webpage
  34. Yohann DE CASTRO 28 One Spike Detection in Sparse Decomposition

    One Spike Model Observation t 2 (R \ Z)d <latexit sha1_base64="7Z9DCOrZqGh02BkZTDr9yxcfxgk=">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</latexit> Z = ⇢(· t) + F?(") <latexit sha1_base64="DaM+kCUm3eliZm2/iPM3UekQAiU=">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</latexit> ⌫0 = t <latexit sha1_base64="f66k3FCNySOK/udcHpNnFnym780=">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</latexit> with Likelihood b t = arg max t |Z(t)|2 <latexit sha1_base64="locHNZfQewWMnlVPOdo1+crevhU=">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</latexit>
  35. Yohann DE CASTRO 28 One Spike Detection in Sparse Decomposition

    One Spike Model Observation t 2 (R \ Z)d <latexit sha1_base64="7Z9DCOrZqGh02BkZTDr9yxcfxgk=">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</latexit> Z = ⇢(· t) + F?(") <latexit sha1_base64="DaM+kCUm3eliZm2/iPM3UekQAiU=">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</latexit> ⌫0 = t <latexit sha1_base64="f66k3FCNySOK/udcHpNnFnym780=">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</latexit> with Likelihood b t = arg max t |Z(t)|2 <latexit sha1_base64="locHNZfQewWMnlVPOdo1+crevhU=">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</latexit> Information Bound: Computational Bound: exponentially large number of critical points close to the maximum ?? b t !n!1 t , = O(??) <latexit sha1_base64="Eeol2K3FK01UdLJpvLbpgJ6GJS8=">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</latexit> = O(n??) <latexit sha1_base64="ly62MpP6skEDBBHsJcS82m9rkaE=">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</latexit>
  36. Yohann DE CASTRO 29 Gaussian widths of Sum-Of-Squares cones Sparse

    Deconvolution and Spiked Tensor PCA involves maxima of polynomials p? = sup x2X f(x) <latexit sha1_base64="ExxQxVUNfpsJUQZ1O7n/HIFkHDc=">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</latexit>
  37. Yohann DE CASTRO 29 Gaussian widths of Sum-Of-Squares cones Sparse

    Deconvolution and Spiked Tensor PCA involves maxima of polynomials p? = sup x2X f(x) <latexit sha1_base64="ExxQxVUNfpsJUQZ1O7n/HIFkHDc=">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</latexit> = sup µ 0, R dµ=1 Z X fdµ <latexit sha1_base64="PqirBvsSPOmqTPCLCZLVOt7E5QY=">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</latexit>
  38. Yohann DE CASTRO 29 Gaussian widths of Sum-Of-Squares cones Sparse

    Deconvolution and Spiked Tensor PCA involves maxima of polynomials p? = sup x2X f(x) <latexit sha1_base64="ExxQxVUNfpsJUQZ1O7n/HIFkHDc=">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</latexit> = sup µ 0, R dµ=1 Z X fdµ <latexit sha1_base64="PqirBvsSPOmqTPCLCZLVOt7E5QY=">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</latexit> M(X) <latexit sha1_base64="+mCJiPo/4fJTvbpMr0pnjFjCZLM=">AAAC23icjVHLSsNAFD2Nr1pfUcGNm2ARdFOSKuiy6MaNUME+oBaZjKMG8yKZCKW6cidu/QG3+j/iH+hfeGdMRS2iE5KcOfeeM3PvdWPfS6VtvxSMkdGx8YniZGlqemZ2zpxfaKZRlnDR4JEfJW2XpcL3QtGQnvRFO04EC1xftNyLXRVvXYok9aLwUPZi0Q3YWeidepxJoo7NpaOAyXPOfGt/7RO214/Nsl2x9bKGgZODMvJVj8xnHOEEETgyBBAIIQn7YEjp6cCBjZi4LvrEJYQ8HRe4Rom0GWUJymDEXtD3jHadnA1przxTreZ0ik9vQkoLq6SJKC8hrE6zdDzTzor9zbuvPdXdevR3c6+AWIlzYv/SDTL/q1O1SJxiW9fgUU2xZlR1PHfJdFfUza0vVUlyiIlT+ITiCWGulYM+W1qT6tpVb5mOv+pMxao9z3MzvKlb0oCdn+McBs1qxdmoVA82y7WdfNRFLGMFazTPLdSwhzoa5H2FBzziyegaN8atcfeRahRyzSK+LeP+HXNpl64=</latexit> with cone of moments and f(x) = X ↵ x↵ <latexit sha1_base64="qug1a45qbf4r1jRdXVSeaU2GXRE=">AAAC33icjVHLSsNAFD3Gd31V3ekmWIS6KWkVdCMU3bhUsLbQ1jJJpxqaF8lEWorgzp249Qfc6t+If6B/4Z1xCmoRnZDkzLn3nJl7rx15biIs63XMGJ+YnJqemc3MzS8sLmWXV86SMI0dXnFCL4xrNku45wa8Ilzh8VoUc+bbHq/a3UMZr17xOHHD4FT0I9702UXgdlyHCaJa2bVOvre130hSv9VgXnTJzN65Bq1szipYapmjoKhBDnodh9kXNNBGCAcpfHAEEIQ9MCT01FGEhYi4JgbExYRcFee4Roa0KWVxymDEdul7Qbu6ZgPaS89EqR06xaM3JqWJTdKElBcTlqeZKp4qZ8n+5j1QnvJuffrb2ssnVuCS2L90w8z/6mQtAh3sqRpcqilSjKzO0S6p6oq8ufmlKkEOEXEStykeE3aUcthnU2kSVbvsLVPxN5UpWbl3dG6Kd3lLGnDx5zhHwVmpUNwulE52cuUDPeoZrGMDeZrnLso4wjEq5H2DRzzh2WDGrXFn3H+mGmNas4pvy3j4ADZ7mYo=</latexit> = sup (y↵)↵ 2M(X), y0=1 X ↵ f↵y↵ <latexit sha1_base64="A8LvG+sThPZ0rfv1QPaymGJVfro=">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</latexit>
  39. Yohann DE CASTRO 30 Gaussian widths of Sum-Of-Squares cones p?

    r = sup (y↵)↵ 2MSDP SoS(r) (X), y0=1 X ↵ f↵y↵ <latexit sha1_base64="XRQfH49QxQVW+5jCzqJyoTx2smA=">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</latexit> In practice we rather consider
  40. Yohann DE CASTRO 30 Gaussian widths of Sum-Of-Squares cones p?

    r = sup (y↵)↵ 2MSDP SoS(r) (X), y0=1 X ↵ f↵y↵ <latexit sha1_base64="XRQfH49QxQVW+5jCzqJyoTx2smA=">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</latexit> In practice we rather consider with 9A1, . . . , Ar psd matrices s.t. <latexit sha1_base64="4kqn7uWCngDM4aUF6UOeoLyZvIU=">AAAC/XicjVHLSuRAFD3G97t1lm4KG8FFExIVdKnjxqWCrYKRJqkutTAvqiqiNI1/MrvZDW79AbfOVuYPxr/wVhnBGRGtkOTUufecqntvUqZSmyD4M+ANDg2PjI6NT0xOTc/MNubmD3RRKS7avEgLdZTEWqQyF20jTSqOSiXiLEnFYXKxbeOHl0JpWeT75roUJ1l8lstTyWNDVKexEYkrOkWzrU7YitJuYXRrq6MiI65Mj5W6y7LYKMmF7kct9kJr3/j9TqMZ+IFb7D0Ia9BEvXaLxiMidFGAo0IGgRyGcIoYmp5jhAhQEneCHnGKkHRxgT4mSFtRlqCMmNgL+p7R7rhmc9pbT+3UnE5J6VWkZFgiTUF5irA9jbl45Zwt+5F3z3nau13TP6m9MmINzon9TPea+VWdrcXgFBuuBkk1lY6x1fHapXJdsTdnb6oy5FASZ3GX4oowd8rXPjOn0a5229vYxf+6TMvaPa9zKzzZW9KAw//H+R4crPjhqr+yt9bc/F6PegwLWMQyzXMdm9jBLtrk/QP3eMBv78b76f3ybl9SvYFa8w3/LO/uGcO+pjU=</latexit> r X ↵=1 y↵A↵ is psd <latexit sha1_base64="UbtgZrqtutwg0d1aEkNmCFYzNe8=">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</latexit> (y↵) 2 MSDP SoS(r) (X) , (y↵) 2 Rn <latexit sha1_base64="EA2ypTHjfx+OV9qVHIKh3z9HwNY=">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</latexit> and <latexit sha1_base64="+bKqGdCUdby4QVFWqCw4G4zxBi4=">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</latexit> 9 yn+1, . . . , yr s.t. <latexit sha1_base64="43ohDNGbyFctNgGv/yFDX9nAMn8=">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</latexit>
  41. Yohann DE CASTRO 30 Gaussian widths of Sum-Of-Squares cones p?

    r = sup (y↵)↵ 2MSDP SoS(r) (X), y0=1 X ↵ f↵y↵ <latexit sha1_base64="XRQfH49QxQVW+5jCzqJyoTx2smA=">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</latexit> In practice we rather consider with 9A1, . . . , Ar psd matrices s.t. <latexit sha1_base64="4kqn7uWCngDM4aUF6UOeoLyZvIU=">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</latexit> r X ↵=1 y↵A↵ is psd <latexit sha1_base64="UbtgZrqtutwg0d1aEkNmCFYzNe8=">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</latexit> (y↵) 2 MSDP SoS(r) (X) , (y↵) 2 Rn <latexit sha1_base64="EA2ypTHjfx+OV9qVHIKh3z9HwNY=">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</latexit> and <latexit sha1_base64="+bKqGdCUdby4QVFWqCw4G4zxBi4=">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</latexit> 9 yn+1, . . . , yr s.t. <latexit sha1_base64="43ohDNGbyFctNgGv/yFDX9nAMn8=">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</latexit> Questions: E(p? r ) when f↵ ⇠i.i.d. N(0, 1) ? <latexit sha1_base64="fqE+uJu0jkMw2bKJo1Yx3fQhnL8=">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</latexit> E(p? r p?) when f↵ ⇠i.i.d. N(0, 1) ? <latexit sha1_base64="9lqXvecCqb7OOOXu1wMq5a0NWWo=">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</latexit>