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Linear Convergence of a Proximal Alternating Minimization Method with Extrapolation for L1-Norm Principal Component Analysis

A popular robust alternative of the classic principal component analysis (PCA) is the
-norm PCA (L1-PCA), which aims to find a subspace that captures the most variation in a dataset as measured by the -norm. L1-PCA has shown great promise in alleviating the effect of outliers in data analytic applications. However, it gives rise to a challenging nonsmooth, nonconvex optimization problem, for which existing algorithms are either not scalable or lack strong theoretical guarantees on their convergence behavior. In this paper, we propose a proximal alternating minimization method with extrapolation (PAMe) for solving a two-block reformulation of the L1-PCA problem. We then show that for both the L1-PCA problem and its two-block reformulation, the Kurdyka–Łojasiewicz exponent at any of the limiting critical points is . This allows us to establish the linear convergence of the sequence of iterates generated by PAMe and to determine the criticality of the limit of the sequence with respect to both the L1-PCA problem and its two-block reformulation. To complement our theoretical development, we show via numerical experiments on both synthetic and real-world datasets that PAMe is competitive with a host of existing methods. Our results not only significantly advance the convergence theory of iterative methods for L1-PCA but also demonstrate the potential of our proposed method in applications.

Ketersediaan

Informasi Detail

Judul Seri

SIAM JOURNAL ON OPTIMIZATION; Vol.33 No.2 April-June 2023

No. Panggil

-

Penerbit

: .,

Deskripsi Fisik

p. 684-712

Bahasa

English

ISBN/ISSN

-

Klasifikasi

NONE

Tipe Isi

-

Tipe Media

-

Tipe Pembawa

-

Edisi

-

Subjek

EXTRAPOLATION
LINEAR CONVERGENCE
L1-PCA
PROXIMAL ALTERNATING MINIMIZATION

Info Detail Spesifik

https://doi.org/10.1137/21M1434507

Pernyataan Tanggungjawab

Peng Wang, Huikang Liu, Anthony Man-Cho So

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