Identifying Outliers in Large Matrices via Randomized Adaptive Compressive Sampling

TL;DR

This paper introduces an adaptive sampling method to efficiently identify outlier columns in large low-rank matrices, with theoretical guarantees and practical applications in collaborative filtering and computer vision.

Contribution

It proposes a novel two-step adaptive sensing approach with proven performance guarantees for outlier detection in large matrices.

Findings

Accurate outlier identification with minimal linear summaries.

Effective in noisy and incomplete data scenarios.

Validated through applications in collaborative filtering and saliency map estimation.

Abstract

This paper examines the problem of locating outlier columns in a large, otherwise low-rank, matrix. We propose a simple two-step adaptive sensing and inference approach and establish theoretical guarantees for its performance; our results show that accurate outlier identification is achievable using very few linear summaries of the original data matrix -- as few as the squared rank of the low-rank component plus the number of outliers, times constant and logarithmic factors. We demonstrate the performance of our approach experimentally in two stylized applications, one motivated by robust collaborative filtering tasks, and the other by saliency map estimation tasks arising in computer vision and automated surveillance, and also investigate extensions to settings where the data are noisy, or possibly incomplete.