Projected Landweber iteration for matrix completion
type: Item
instance of: scholarly article
title: Projected Landweber iteration for matrix completion
zbMATH Open document ID: 1225.65049
published in: Journal of Computational and Applied Mathematics
publication date: 2010-10-11T00:00:00Z
review text: Let \(M\in\mathbb{R}^{n\times m}\) be an unknown low-rank matrix and \(\{M_{ij}:\;(i,j)\in\Omega\}\) be a sampling set of its entries, and let \(\|\cdot \|_F\), \(\|\cdot\|_*\) be the Frobenius norm and the nuclear norm on \(\mathbb{R}^{n\times m}\), respectively. The authors derive a nonlinear constrained quadratic programming problem: \(\min\{\|\mathcal{P}_\Omega(X)-\mathcal{P}_\Omega(M)\|_F^2:\;X\in \bar{B}_R\}\), where \(\mathcal{P}_\Omega\) is the orthogonal projector such that \(\mathcal{P}_\Omega(X)_{ij}=M_{ij}\) when \((i,j)\in\Omega\) and otherwise equals to zero, and the level set \(\bar{B}_R=\{X\in\mathbb{R}^{n\times m}: \|X\|_*\leq R\}\). They design a new algorithm for addressing such a matrix completion problem, namely projected Landweber iteration, based on the projected method and Landweber iteration. The convergence of the proposed algorithm is proved strictly, and numerical results show that the proposed algorithm can be fast and efficient under suitable prior conditions of \(M\).
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 15A83
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 90C20
zbMATH DE Number: 5798183
zbMATH Keywords: matrix completion
zbMATH Keywords: projected Landweber iteration
zbMATH Keywords: nuclear norm
zbMATH Keywords: optimization
zbMATH Keywords: Frobenius norm
zbMATH Keywords: nonlinear constrained quadratic programming
zbMATH Keywords: algorithm
zbMATH Keywords: convergence
zbMATH Keywords: numerical results
author: Hui Zhang
author: Li-zhi Cheng
describes a project that uses: SPGL1
describes a project that uses: SDPT3
describes a project that uses: PROPACK
describes a project that uses: ParNes
describes a project that uses: PDCO
describes a project that uses: Eigentaste
MaRDI profile type: Publication
full work available at URL: j.cam.2010.06.010
OpenAlex ID: W1984874910
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DOI: 10.1016/J.CAM.2010.06.010
description: scientific article; zbMATH DE number 5798183