Low-rank matrix completion and denoising under Poisson noise

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Andrew D McRae;Mark A Davenport
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引用次数: 12

Abstract

This paper considers the problem of estimating a low-rank matrix from the observation of all or a subset of its entries in the presence of Poisson noise. When we observe all entries, this is a problem of matrix denoising; when we observe only a subset of the entries, this is a problem of matrix completion. In both cases, we exploit an assumption that the underlying matrix is low-rank. Specifically, we analyse several estimators, including a constrained nuclear-norm minimization program, nuclear-norm regularized least squares and a non-convex constrained low-rank optimization problem. We show that for all three estimators, with high probability, we have an upper error bound (in the Frobenius norm error metric) that depends on the matrix rank, the fraction of the elements observed and the maximal row and column sums of the true matrix. We furthermore show that the above results are minimax optimal (within a universal constant) in classes of matrices with low-rank and bounded row and column sums. We also extend these results to handle the case of matrix multinomial denoising and completion.
泊松噪声下的低秩矩阵补全与去噪
本文考虑了在存在泊松噪声的情况下,根据对其所有或子集项的观测来估计低秩矩阵的问题。当我们观察所有条目时,这是一个矩阵去噪的问题;当我们只观察条目的子集时,这是一个矩阵完备的问题。在这两种情况下,我们都利用了一个假设,即底层矩阵是低秩的。具体地,我们分析了几个估计量,包括约束核范数最小化程序、核范数正则化最小二乘和非凸约束低秩优化问题。我们证明,对于所有三个估计量,在高概率的情况下,我们有一个误差上界(在Frobenius范数误差度量中),它取决于矩阵秩、观察到的元素的分数以及真矩阵的最大行和和列和。我们进一步证明了在具有低秩和有界行和列和的矩阵类中,上述结果是极小极大最优的(在通用常数内)。我们还将这些结果扩展到处理矩阵多项式去噪和补全的情况。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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