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High Dimensional Low Rank Plus Sparse Matrix Decomposition

机译：高维低秩加稀疏矩阵分解

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摘要

This paper is concerned with the problem of low-rank plus sparse matrix decomposition for big data. Conventional algorithms for matrix decomposition use the entire data to extract the low-rank and sparse components, and are based on optimization problems with complexity that scales with the dimension of the data, which limits their scalability. Furthermore, existing randomized approaches mostly rely on uniform random sampling, which is quite inefficient for many real world data matrices that exhibit additional structures (e.g., clustering). In this paper, a scalable subspace-pursuit approach that transforms the decomposition problem to a subspace learning problem is proposed. The decomposition is carried out by using a small data sketch formed from sampled columns/rows. Even when the data are sampled uniformly at random, it is shown that the sufficient number of sampled columns/rows is roughly O(rμ), where μ is the coherency parameter and r is the rank of the low-rank component. In addition, adaptive sampling algorithms are proposed to address the problem of columns/rows sampling from structured data. We provide an analysis of the proposed method with adaptive sampling and show that adaptive sampling makes the required number of sampled columns/rows invariant to the distribution of the data. The proposed approach is amenable to online implementation and an online scheme is proposed.
机译：本文涉及大数据的低秩加稀疏矩阵分解问题。用于矩阵分解的常规算法使用整个数据来提取低秩和稀疏分量，并且基于复杂度随数据维度缩放的优化问题，从而限制了它们的可伸缩性。此外，现有的随机方法大多依赖于统一的随机采样，这对于表现出额外结构（例如，聚类）的许多现实世界数据矩阵而言是相当低效的。本文提出了一种可扩展的子空间追求方法，将分解问题转化为子空间学习问题。通过使用由采样的列/行组成的小数据草图进行分解。即使当随机地对数据进行均匀采样时，也显示出足够的采样列数/行数大致为O（rμ），其中μ是相干性参数，r是低秩分量的秩。另外，提出了自适应采样算法以解决来自结构化数据的列/行采样的问题。我们通过自适应采样对提出的方法进行了分析，结果表明自适应采样使所需的采样列数/行数不随数据的分布而变化。所提出的方法适合在线实施并且提出了在线方案。

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