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 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 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, K

Mots clés: asymptotically level stable functions; global minimizers; local minimizers; ℓ0 regularization; nonconvex nonsmooth minimization; perturbation analysis; quadratic programming; solution analysis; sparse recovery; strict minimizers; under-determined linear systems; uniqueness of the solution; variational methods

Domaines: Analyse numérique [math.NA]

Mila Nikolova. Description of the minimizers of least squares regularized with ℓ0-norm. Uniqueness of the global minimizer. 2013. ⟨hal-00723812v4⟩