Optimal Shrinkage of Singular Values Under Random Data Contamination

TL;DR

This paper introduces a unified framework for handling various data contamination models in low-rank matrix reconstruction, developing an optimal algorithm based on singular value manipulation and identifying a fundamental signal-to-noise threshold.

Contribution

It presents an asymptotically optimal method for low-rank matrix recovery under diverse contamination models using singular value adjustments.

Findings

Unified framework for contamination models

Optimal singular value-based reconstruction algorithm

Explicit signal-to-noise cutoff for successful estimation

Abstract

A low rank matrix X has been contaminated by uniformly distributed noise, missing values, outliers and corrupt entries. Reconstruction of X from the singular values and singular vectors of the contaminated matrix Y is a key problem in machine learning, computer vision and data science. In this paper we show that common contamination models (including arbitrary combinations of uniform noise,missing values, outliers and corrupt entries) can be described efficiently using a single framework. We develop an asymptotically optimal algorithm that estimates X by manipulation of the singular values of Y , which applies to any of the contamination models considered. Finally, we find an explicit signal-to-noise cutoff, below which estimation of X from the singular value decomposition of Y must fail, in a well-defined sense.