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Reweighted Quasi Norm Regularized Low-Rank Factorization for Matrix Robust PCA

Zhenzhi Qin, Liping Zhang

TL;DR

A novel strategy in non-convex quasi-norm representation is proposed, introducing a method to obtain weighted matrix quasi-norm factorization forms.

Abstract

Robust Principal Component Analysis (RPCA) and its associated non-convex relaxation methods constitute a significant component of matrix completion problems, wherein matrix factorization strategies effectively reduce dimensionality and enhance computational speed. However, some non-convex factorization forms lack theoretical guarantees. This paper proposes a novel strategy in non-convex quasi-norm representation, introducing a method to obtain weighted matrix quasi-norm factorization forms. Especially, explicit bilinear factor matrix factorization formulations for the weighted logarithmic norm and weighted Schatten-$q$ quasi norms with $q=1, 1/2, 2/3$ are provided, along with the establishment of corresponding matrix completion models. An Alternating Direction Method of Multipliers (ADMM) framework algorithm is employed for solving, and convergence results of the algorithm are presented.

Reweighted Quasi Norm Regularized Low-Rank Factorization for Matrix Robust PCA

TL;DR

A novel strategy in non-convex quasi-norm representation is proposed, introducing a method to obtain weighted matrix quasi-norm factorization forms.

Abstract

Robust Principal Component Analysis (RPCA) and its associated non-convex relaxation methods constitute a significant component of matrix completion problems, wherein matrix factorization strategies effectively reduce dimensionality and enhance computational speed. However, some non-convex factorization forms lack theoretical guarantees. This paper proposes a novel strategy in non-convex quasi-norm representation, introducing a method to obtain weighted matrix quasi-norm factorization forms. Especially, explicit bilinear factor matrix factorization formulations for the weighted logarithmic norm and weighted Schatten- quasi norms with are provided, along with the establishment of corresponding matrix completion models. An Alternating Direction Method of Multipliers (ADMM) framework algorithm is employed for solving, and convergence results of the algorithm are presented.
Paper Structure (5 sections, 14 theorems, 94 equations, 2 algorithms)

This paper contains 5 sections, 14 theorems, 94 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Factorization Strategy
  4. Factorization Forms of Weighted Schatten-q Norms
  5. Algorithms and convergence result

Key Result

Lemma 2.1

Let $\mathcal{M}$ be an embedded submanifold of $\mathbb{R}^{m}$. $F$ is Lipschitz continuous at $x\in\mathcal{M}$, and $\bar{F}\doteq F|_{\mathcal{M}}$. If $F$ is regular along $T_x\mathcal{M}$ at $x \in \mathcal{M}$, then we have $\partial\bar{F}(x)={\rm Proj}_{T_x\mathcal{M}}(\partial F(x))$, whe

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: Riemann manifold
  • Definition 2.5: Chart boumal2023introduction
  • Definition 2.6: Lipschitz Continuity on Manifolds Zhang2013Optimality
  • Definition 2.7: Generalized Clarke Subdifferential Hpsseini2011Generalized
  • Definition 2.8: Definition 5.2 Zhang2013Optimality
  • Lemma 2.1: Theorem 5.1 (iii) Zhang2013Optimality
  • Lemma 2.2: Lemma 5.1 Zhang2013Optimality
  • ...and 23 more