Can critical-point paths under ℓp-regularization (0 < p < 1) reach the sparsest least squares solutions?

Kwangjin Jeong, Masahiro Yukawa, Shun Ichi Amari

    Research output: Contribution to journalArticle

    3 Citations (Scopus)

    Abstract

    The solution path of the least square problem under ℓp- regularization (0<p<1) is studied, where the Lagrangian multiplier λ due to the constraint is the parameter of the path. It is first proven that the least square solution of an unconstrained overdetermined linear system is connected with the origin, under a mild condition, by a continuous path of critical points of an p-regularized squared error function. Based on this fact, it is proven that every sparsest least square solution of an underdetermined system is connected with the origin by a critical-point path. The existence theorem holds more generally for any least square solution whose support has its associated submatrix of the fat sensing matrix be full column rank. This is a sufficient condition for the existence, and allows to reduce the underdetermined problem to an overdetermined one with the off-support variable(s) nullified. A necessary condition is that the gradient of the ℓp regularizer with respect to the support variables lies in the row space of the submatrix (which is not necessarily full column rank).

    Original languageEnglish
    Article number6776531
    Pages (from-to)2960-2968
    Number of pages9
    JournalIEEE Transactions on Information Theory
    Volume60
    Issue number5
    DOIs
    Publication statusPublished - 2014 May

    ASJC Scopus subject areas

    • Information Systems
    • Computer Science Applications
    • Library and Information Sciences

    Fingerprint Dive into the research topics of 'Can critical-point paths under ℓp-regularization (0 < p < 1) reach the sparsest least squares solutions?'. Together they form a unique fingerprint.

  • Cite this