English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we investigate the recovery of the sparse representation ofdata in general infinite-dimensional optimization problems regularized byconvex functionals. We show that it is possible to define a suitablenon-degeneracy condition on the minimal-norm dual certificate, extending thewell-established non-degeneracy source condition (NDSC) associated with totalvariation regularized problems in the space of measures, as introduced in(Duval and Peyr\'e, FoCM, 15:1315-1355, 2015). In our general setting, we needto study how the dual certificate is acting, through the duality product, onthe set of extreme points of the ball of the regularizer, seen as a metricspace. This justifies the name Metric Non-Degenerate Source Condition (MNDSC).More precisely, we impose a second-order condition on the dual certificate,evaluated on curves with values in small neighbourhoods of a given collectionof n extreme points. By assuming the validity of the MNDSC, together with thelinear independence of the measurements on these extreme points, we establishthat, for a suitable choice of regularization parameters and noise levels, theminimizer of the minimization problem is unique and is uniquely represented asa linear combination of n extreme points. The paper concludes by obtainingexplicit formulations of the MNDSC for three problems of interest. First, weexamine total variation regularized deconvolution problems, showing that theclassical NDSC implies our MNDSC, and recovering a result similar to (Duval andPeyr\'e, FoCM, 15:1315-1355, 2015). Then, we consider 1-dimensional BVfunctions regularized with their BV-seminorm and pairs of measures regularizedwith their mutual 1-Wasserstein distance. In each case, we provide explicitversions of the MNDSC and formulate specific sparse representation recoveryresults.

A General Theory for Exact Sparse Representation Recovery in Convex Optimization