Régularisation de problèmes inverses Analyse unifiée de la robustesse Samuel VAITER CNRS, CEREMADE, Université Paris-Dauphine, France Travaux en collaboration avec M. GOLBABAEE, G. PEYRÉ et J. FADILI 

Linear Inverse Problems inpainting denoising super-resolution Forward model y = x0 + w observations noise input operator 

The Variational Approach x argmin x RN 1 2 ||y x||2 2 + J(x) Data fidelity Regularity 

The Variational Approach x argmin x RN 1 2 ||y x||2 2 + J(x) Data fidelity Regularity J(x) = || x||2 J(x) = || x||1 

The Variational Approach x argmin x RN 1 2 ||y x||2 2 + J(x) Data fidelity Regularity J(x) = || x||2 J(x) = || x||1 sparsity analysis-sparsity group-sparsity nuclear norm Tikhonov Total Variation Anti-sparse Polyhedral L1 + TV ... atomic norm decomposable norm Candidate J 

Objectives Model selection performance x0 x w Prior model J 

Objectives Model selection performance x0 x w Prior model J 

Objectives Model selection performance x0 x w Prior model J How close ? in term of SNR in term of features 

Union of Linear Models Union of models: T T linear spaces 

Union of Linear Models Union of models: T T linear spaces T sparsity 

Union of Linear Models Union of models: T T linear spaces block sparsity T sparsity 

Union of Linear Models Union of models: T T linear spaces block sparsity T sparsity analysis sparsity 

Union of Linear Models Union of models: T T linear spaces block sparsity low rank T sparsity analysis sparsity 

Union of Linear Models Union of models: T T linear spaces block sparsity low rank Objective Encode T in a function T sparsity analysis sparsity 

Gauges 1 J(x) J : RN R+ convex J( x) = J(x), 0 

Gauges 1 J(x) J : RN R+ convex C C = {x : J(x) 1} J( x) = J(x), 0 

Gauges 1 J(x) J : RN R+ convex C C = {x : J(x) 1} Geometry of C Union of Models (T )T T x T x x 0 T0 x x 0 T0 x 0 x T T0 ||x||1 |x1|+||x2,3|| ||x|| ||x|| J( x) = J(x), 0 

Subdifferential |x| 0 

Subdifferential |x| 0 

J(x) = RN : x , J(x ) J(x)+ , x x Subdifferential |x| 0 

J(x) = RN : x , J(x ) J(x)+ , x x Subdifferential |x| 0 |·|(0) = [ 1,1] x = 0, |·|(x) = {sign(x)} 

From the Subdifferential to the Model J(x) x 0 J(x) x 0 

From the Subdifferential to the Model J(x) x 0 J(x) x 0 Tx= VectHull( J(x)) Tx Tx Tx = : supp( ) supp(x) 

From the Subdifferential to the Model J(x) x 0 J(x) x 0 ex = ProjTx ( J(x)) ex ex ex = sign(x) Tx= VectHull( J(x)) Tx Tx Tx = : supp( ) supp(x) 

Regularizations and their Models J(x) = ||x||1 ex = sign(x) Tx = : supp( ) supp(x) x x J(x) = b ||xb|| ex = (N (xb))b B Tx = : supp( ) supp(x) x N (xb) = xb/||xb|| J(x) = ||x||∗ ex =UV Tx = : U V = 0 x x =UΛV ∗ J(x) = ||x||∞ ex = |I| 1 sign(x) Tx = : I sign(xI ) x x I = {i : |xi | = ||x||∞} 

Dual Certificates and Model Selection x argmin x RN 1 2 ||y x||2 2 + J(x) Hypothesis: Ker Tx0 = {0} J regular enough 

Dual Certificates and Model Selection x argmin x RN 1 2 ||y x||2 2 + J(x) Hypothesis: Ker Tx0 = {0} J regular enough ¯ D = Im ri( J(x0)) Tight dual certificates: x = x0 J(x) x 

Dual Certificates and Model Selection x argmin x RN 1 2 ||y x||2 2 + J(x) Hypothesis: 0 = ( + Tx0 ) ex0 Minimal norm pre-certificate: Tx = Tx0 and ||x x0|| = O(||w||) If 0 ¯ D,||w|| small enough and ||w||, then x is the unique solution. Moreover, [V. et al. 2013] 1: [Fuchs 2004] 1 2: [Bach 2008] Ker Tx0 = {0} J regular enough ¯ D = Im ri( J(x0)) Tight dual certificates: x = x0 J(x) x 

Example: Sparse Deconvolution x = i xi (· i) J(x) = ||x||1 Increasing : reduces correlation. reduces resolution. x0 x0 

Example: Sparse Deconvolution x = i xi (· i) J(x) = ||x||1 Increasing : reduces correlation. reduces resolution. x0 x0 I = j : x0[j] = 0 || 0,Ic || < 1 0 ¯ D support recovery || 0,Ic || 1 2 20 

Example: 1D TV Denoising J(x) = || x||1 = Id I = {i : ( x0)i = 0} x0 

Example: 1D TV Denoising J(x) = || x||1 = Id I = {i : ( x0)i = 0} x0 +1 1 0 = div( 0) where j I,( 0)j = 0 x0 I J || 0,Ic || < 1 Support stability 

Example: 1D TV Denoising J(x) = || x||1 = Id I = {i : ( x0)i = 0} x0 +1 1 0 = div( 0) where j I,( 0)j = 0 x0 I J || 0,Ic || < 1 Support stability x0 || 0,Ic || = 1 2-stability only 

Conclusion Gauges: encode linear models as singular points 

Conclusion Gauges: encode linear models as singular points Certificates: guarantees of model selection / 2 robustness (see poster 208 for a pure robustness result) 

Conclusion Merci de votre attention ! Gauges: encode linear models as singular points Certificates: guarantees of model selection / 2 robustness (see poster 208 for a pure robustness result)