Iterative Thresholding Meets Free-Discontinuity Problems

Accelerated Projected Gradient Method for Linear Inverse Problems with Sparsity Constraints

Regularization of ill-posed linear inverse problems via l1 penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an l1 penalized functional is via an iterative soft-thresholding algorithm. We propose an alternative implementation to l1-constraints, using a gradient method, with projection on l1-balls. The correspondin...

PArameter-free Generic Iterative Thresholding Algorithm (PAGITA)

Minimization of Transformed L1 Penalty: Closed Form Representation and Iterative Thresholding Algorithms

The transformed l1 penalty (TL1) functions are a one parameter family of bilinear transformations composed with the absolute value function. When acting on vectors, the TL1 penalty interpolates l0 and l1 similar to lp norm (p ∈ (0, 1)). In our companion paper, we showed that TL1 is a robust sparsity promoting penalty in compressed sensing (CS) problems for a broad range of incoherent and cohere...

An Iterative Thresholding Algorithm for Linear Inverse Problems with a Sparsity Constraint

We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted ppenalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem. Use of such p-penalized problems with p < 2 is often advocated when one expects ...

Transformed Schatten-1 Iterative Thresholding Algorithms for Matrix Rank Minimization and Applications

We study a non-convex low-rank promoting penalty function, the transformed Schatten-1 (TS1), and its applications in matrix completion. The TS1 penalty, as a matrix quasi-norm defined on its singular values, interpolates the rank and the nuclear norm through a nonnegative parameter a∈ (0,+∞). We consider the unconstrained TS1 regularized low-rank matrix recovery problem and develop a fixed poin...