Description of the minimizers of least squares regularized with L0-norm. Uniqueness of the global minimizer

Mila Nikolova (1)

1 CMLA - Centre de Mathématiques et de Leurs Applications

We have an M x N real-valued arbitrary matrix A (e.g. a dictionary) with M0. For several decades, this objective has attracted 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 b. Each strict local minimizer is linear in data. It is proved that F has global minimizers and that they are always strict. They are studied in more details under the (standard) assumption that rank(A)=Mb_k, all global minimizers of F are k-sparse. An assumption on A is adopted and proved to fail only on a closed negligible subset. Then for all data d beyond a closed negligible subset, the objective F for b>b_k, k

Mots clés: asymptotically level stable functions; global minimizers; local minimizers; $\ell_0$ regularization; nonconvex nonsmooth minimization; perturbation analysis; quadratic programming; solution analysis; sparse recovery; strict minimizers; underdetermined linear systems; uniqueness of the solution; variational methods

Domaines: Analyse numérique [math.NA]; Traitement du signal et de l'image [eess.SP]

Mila Nikolova. Description of the minimizers of least squares regularized with L0-norm. Uniqueness of the global minimizer. SIAM Journal on Imaging Sciences, 2013, 6 (2), pp.904 - 937. ⟨10.1137/11085476X⟩. ⟨hal-00723812v7⟩