In this talk, we investigate in a unified way the structural properties of a large class of convex regularizers for linear inverse problems. We consider regularizations with convex positively 1-homogenous functionals (so-called gauges) which are piecewise smooth. Singularies of such functionals are crucial to force the solution to the regularization to belong to an union of linear space of low dimension. These spaces (the so-called "models") allows one to encode many priors on the data to be recovered, conforming to some notion of simplicity/low complexity. This family of priors encompasses many special instances routinely used in regularized inverse problems such as L^1, L^1-L^2 (group sparsity), nuclear norm, or the L^infty norm. The piecewise-regular requirement is flexible enough to cope with analysis-type priors that include a pre-composition with a linear operator, such as for instance the total variation and polyhedral gauges. This notion is also stable under summation of regularizers, thus enabling to handle mixed regularizations.

The main set of contributions of this talk is dedicated to assessing the theoretical recovery performance of this class of regularizers. We provide sufficient conditions that allow to provably controlling the deviation of the recovered solution from the true underlying object, as a function of the noise level. More precisely we establish two main results. The first one ensures that the solution to the inverse problem is unique and lives on the same low dimensional sub-space as the true vector to recover, with the proviso that the minimal signal to noise ratio is large enough. This extends previous results well-known for the L^1 norm [1], analysis L^1 semi-norm [2], and the nuclear norm [3] to the general class of piecewise smooth gauges. In the second result, we establish L^2 stability by showing that the L^2 distance between the recovered and true vectors is within a factor of the noise level, thus extending results that hold for coercive convex positively 1-homogenous functionals [4].

This is a joint work with S. Vaiter, C. Deledalle, M. Golbabaee and J. Fadili. For more details, see [5].

Bibliography:

[1] J.J. Fuchs, On sparse representations in arbitrary
redundant bases. IEEE Transactions on Information Theory, 50(6):1341-1344,
2004.

[2] S. Vaiter, G. Peyré, C. Dossal, J. Fadili, Robust
Sparse Analysis Regularization, to appear in IEEE Transactions on Information
Theory, 2013.

[3] F. Bach, Consistency of trace norm minimization,
Journal of Machine Learning Research, 9, 1019-1048, 2008.

[4] M. Grasmair, Linear convergence rates for Tikhonov
regularization with positively homogeneous functionals. Inverse Problems,
27(7):075014, 2011.

[5] S. Vaiter, M. Golbabaee, J. Fadili, G. Peyré, Model
Selection with Piecewise Regular Gauges, Preprint hal-00842603, 2013