Description of the minimizers of least squares regularized with ℓ0-norm. Uniqueness of the global minimizer
Résumé
We have an M×N real-valued arbitrary matrix A (e. g. a dictionary) with M < N and data d describing the sought-after object with the help of A. This work provides an in-depth analysis of the (local and global) minimizers of an objective function F combining a quadratic data-fidelity term and an ℓ0 penalty applied to each entry of the sought after solution, weighted by a regularization parameter β > 0. For several decades, this objective focuses a ceaseless effort to conceive algorithms approaching a good minimizer. Our theoretical contributions, summarized below, shed new light on the existing algorithms and can help the conception of innovative numerical schemes. To solve the normal equation associated with any M-row submatrix of A is equivalent to compute a local minimizer u* of F. (Local) minimizers u* of F are strict if and only if the submatrix, composed of those columns of A whose indexes form the support of u*, has full column rank. An outcome is that strict local minimizers of F are easily computed without knowing the value of β. Each strict local minimizer is linear in data. The global minimizers of F are always strict. They are studied in more details under the (standard) assumption that rank(A) = M < N. The global minimizers with M-length support are seen to be impractical. Given d, critical values β(K) for any K < M are exhibited such that if β > β(K), all global minimizers of F are K-sparse. An assumption on A is adopted and proven to be generically true. Then for generically all data d, the objective F has a unique global minimizer and the latter is K-sparse for K < M. Instructive small-size (5 × 10) numerical illustrations confirm the main theoretical results.
Domaines
Analyse numérique [math.NA]
Origine : Fichiers produits par l'(les) auteur(s)