Inertial proximal alternating linearized minimization (iPALM) for nonconvex and nonsmooth problems

Thomas Pock, Shoham Sabach

Publikation: Beitrag in einer FachzeitschriftArtikelForschungBegutachtung

Abstract

In this paper we study nonconvex and nonsmooth optimization problems with semialgebraic data, where the variables vector is split into several blocks of variables. The problem consists of one smooth function of the entire variables vector and the sum of nonsmooth functions for each block separately. We analyze an inertial version of the proximal alternating linearized minimization algorithm and prove its global convergence to a critical point of the objective function at hand. We illustrate our theoretical findings by presenting numerical experiments on blind image deconvolution, on sparse nonnegative matrix factorization and on dictionary learning, which demonstrate the viability and effectiveness of the proposed method.

Originalspracheenglisch
Seiten (von - bis)1756-1787
Seitenumfang32
FachzeitschriftSIAM Journal on Imaging Sciences
Jahrgang9
Ausgabenummer4
DOIs
PublikationsstatusVeröffentlicht - 2016

Fingerprint

Nonsmooth Function
Non-negative Matrix Factorization
Nonsmooth Optimization
Nonconvex Optimization
Deconvolution
Sparse matrix
Glossaries
Factorization
Viability
Global Convergence
Smooth function
Critical point
Objective function
Numerical Experiment
Entire
Optimization Problem
Demonstrate
Experiments
Learning
Dictionary

Schlagwörter

    ASJC Scopus subject areas

    • !!Mathematics(all)
    • Angewandte Mathematik

    Dies zitieren

    Inertial proximal alternating linearized minimization (iPALM) for nonconvex and nonsmooth problems. / Pock, Thomas; Sabach, Shoham.

    in: SIAM Journal on Imaging Sciences, Jahrgang 9, Nr. 4, 2016, S. 1756-1787.

    Publikation: Beitrag in einer FachzeitschriftArtikelForschungBegutachtung

    @article{83bdf0d6816942cc881c572090645113,
    title = "Inertial proximal alternating linearized minimization (iPALM) for nonconvex and nonsmooth problems",
    abstract = "In this paper we study nonconvex and nonsmooth optimization problems with semialgebraic data, where the variables vector is split into several blocks of variables. The problem consists of one smooth function of the entire variables vector and the sum of nonsmooth functions for each block separately. We analyze an inertial version of the proximal alternating linearized minimization algorithm and prove its global convergence to a critical point of the objective function at hand. We illustrate our theoretical findings by presenting numerical experiments on blind image deconvolution, on sparse nonnegative matrix factorization and on dictionary learning, which demonstrate the viability and effectiveness of the proposed method.",
    keywords = "Alternating minimization, Blind image deconvolution, Block coordinate descent, Dictionary learning, Heavy ball method, Kurdyka-Łojasiewicz property, Nonconvex and nonsmooth minimization, Sparse nonnegative matrix factorization",
    author = "Thomas Pock and Shoham Sabach",
    year = "2016",
    doi = "10.1137/16M1064064",
    language = "English",
    volume = "9",
    pages = "1756--1787",
    journal = "SIAM journal on imaging sciences",
    issn = "1936-4954",
    publisher = "Society for Industrial and Applied Mathematics Publications",
    number = "4",

    }

    TY - JOUR

    T1 - Inertial proximal alternating linearized minimization (iPALM) for nonconvex and nonsmooth problems

    AU - Pock, Thomas

    AU - Sabach, Shoham

    PY - 2016

    Y1 - 2016

    N2 - In this paper we study nonconvex and nonsmooth optimization problems with semialgebraic data, where the variables vector is split into several blocks of variables. The problem consists of one smooth function of the entire variables vector and the sum of nonsmooth functions for each block separately. We analyze an inertial version of the proximal alternating linearized minimization algorithm and prove its global convergence to a critical point of the objective function at hand. We illustrate our theoretical findings by presenting numerical experiments on blind image deconvolution, on sparse nonnegative matrix factorization and on dictionary learning, which demonstrate the viability and effectiveness of the proposed method.

    AB - In this paper we study nonconvex and nonsmooth optimization problems with semialgebraic data, where the variables vector is split into several blocks of variables. The problem consists of one smooth function of the entire variables vector and the sum of nonsmooth functions for each block separately. We analyze an inertial version of the proximal alternating linearized minimization algorithm and prove its global convergence to a critical point of the objective function at hand. We illustrate our theoretical findings by presenting numerical experiments on blind image deconvolution, on sparse nonnegative matrix factorization and on dictionary learning, which demonstrate the viability and effectiveness of the proposed method.

    KW - Alternating minimization

    KW - Blind image deconvolution

    KW - Block coordinate descent

    KW - Dictionary learning

    KW - Heavy ball method

    KW - Kurdyka-Łojasiewicz property

    KW - Nonconvex and nonsmooth minimization

    KW - Sparse nonnegative matrix factorization

    UR - http://www.scopus.com/inward/record.url?scp=85007346120&partnerID=8YFLogxK

    U2 - 10.1137/16M1064064

    DO - 10.1137/16M1064064

    M3 - Article

    VL - 9

    SP - 1756

    EP - 1787

    JO - SIAM journal on imaging sciences

    JF - SIAM journal on imaging sciences

    SN - 1936-4954

    IS - 4

    ER -