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.
. 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
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
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
t1 <latexit sha1_base64="cWCKPaqr1nM1vHGBVw2a+ALpExw=">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</latexit> One Spike Model
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=">AAAC+3icjVFdaxQxFD2d+tHWj67toy/BRdgiLjOrqC/CoiA+ruC2xU5dMtm0G5qZDElmoZT+k771TfrqH/DVvov/QP+FN3EKahHNMDMn595zkntvUWvlfJp+XUgWr1y9dn1peeXGzVu3Vzt31jadaayQY2G0sdsFd1KrSo698lpu11bystByqzh4GeJbc2mdMtVbf1jL3ZLvV2pPCe6JmnSevGPPGZ9kuZ2ZXi6mxj/0k2yDPWB5yf1McM1evc+d57aXz7mVtVPaVBuTTjftp3GxyyBrQRftGpnOF+SYwkCgQQmJCp6wBoejZwcZUtTE7eKIOEtIxbjEMVZI21CWpAxO7AF992m307IV7YOni2pBp2h6LSkZ7pPGUJ4lHE5jMd5E58D+zfsoeoa7HdK/aL1KYj1mxP5Ld5H5v7pQi8censUaFNVURyZUJ1qXJnYl3Jz9UpUnh5q4gKcUt4RFVF70mUWNi7WH3vIY/xYzAxv2os1t8D3ckgac/TnOy2Bz0M8e9QdvHneHL9pRL+Eu7qFH83yKIV5jhDF5n+ATPuM8OU5Okw/J2c/UZKHVrOO3lXz8ATRzo6w=</latexit>
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=">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</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>
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
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
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),
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
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
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>
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>
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>
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=">AAADCXicjVHNThsxGJwsbUmhLWk59mI1QuqhinYDUrkgRbQHjkEiJBKLIq/jBCv7V6+3UhTlCXgTbtxQr32B3hA8AX2LfjaLRIsq8Gp3x/PNjP3ZUR6rwvj+Vc1bevb8xXL95crqq9dv1hpv3x0WWamF7IkszvQg4oWMVSp7RplYDnIteRLFsh9Nv9h6/7vUhcrSAzPL5XHCJ6kaK8ENUcPG152wKPPhPEzKcCK/Mf8TC1VqwoSbE52wEfE7wcJSVkOk4DEbLNj4vmLYaPot3w32EAQVaKIa3axxiRAjZBAokUAihSEcg6Og5wgBfOTEHWNOnCakXF1igRXylqSSpODETuk7odlRxaY0t5mFcwtaJaZXk5NhgzwZ6TRhuxpz9dIlW/Z/2XOXafc2o39UZSXEGpwQ+5jvTvlUn+3FYIxt14OinnLH2O5ElVK6U7E7Z/e6MpSQE2fxiOqasHDOu3NmzlO43u3Zcle/cUrL2rmotCV+213SBQf/XudDcNhuBZut9v5Ws7NbXXUd7/EBH+k+P6ODPXTRo+wz/MIVrr1T79y78H7cSr1a5VnHX8P7+QdH6qso</latexit> M(X) <latexit sha1_base64="+mCJiPo/4fJTvbpMr0pnjFjCZLM=">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</latexit> with cone of moments and f(x) = X ↵ x↵ <latexit sha1_base64="qug1a45qbf4r1jRdXVSeaU2GXRE=">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</latexit> = sup (y↵)↵ 2M(X), y0=1 X ↵ f↵y↵ <latexit sha1_base64="A8LvG+sThPZ0rfv1QPaymGJVfro=">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</latexit>
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
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>
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>
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ykaneko1992
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