SURE-based Parameter Selection for Sparse Regularization of Inverse Problems

Gabriel Peyre
Université de Paris IX (Paris-Dauphine)

In this talk, I will reviews both theoretical and numerical aspects of parameter selection for inverse problems regularization. We focus our attention to a set of methods built on top of the Generalized Stein Unbiased Risk Estimator (GSURE) [1]. GSURE allows one to estimate without bias the squared error on the orthogonal of the kernel of the imaging operator. One can thus automatically set the value of some parameters of the method by minimizing the GSURE. Computing the GSURE necessitates the estimation of the generalized degree of freedom of the method. We prove in [2] a formula that gives an unbiased estimator of this degree of freedom for sparse l1 analysis regularization. This includes analysis-type translation invariant wavelet sparsity and total variation. This theoretical analysis provides a better understanding of the behavior of the methods, but is difficult to compute numerically for large scale imaging problems. Indeed, convex optimization solvers only provide an approximate solution, which does not lead to a stable estimation of the number of degrees of freedom. We addressed this issue in [3] by proposing a novel algorithm that computes a unbiased and stable estimator of the risk associated to each iterate of a large class of convex optimization methods. The algorithms that I will present can be implemented and tested via the "Numerical Tours" plateform that is available online from This is a joint work with Samuel Vaiter, Charles Deledalle, Jalal Fadili and Charles Dossal. Bibliography: [1] Y. C. Eldar, “Generalized sure for exponential families: Applications to regularization,” IEEE Transactions on Signal Processing, vol. 57, pp. 471– 481, 2009. [2] S. Vaiter, C. Deledalle, G. Peyre, J. Fadili, and C. Dossal, “Local be- havior of sparse analysis regularization: Applications to risk estimation,” Technical report, Preprint Hal-00687751, 2012. [3] C. Deledalle, S. Vaiter, G. Peyre, J. Fadili, and C. Dossal, “Proximal splitting derivatives for risk estimation,” Proc. NCMIP’12, 2012.

Presentation (PDF File)

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