# Low Rank Matrix Recovery with Simultaneous Presence of Outliers and Sparse Corruption

## TL;DR

This paper introduces a robust PCA algorithm capable of simultaneously handling element-wise sparse corruptions and outlier columns in data matrices, with a scalable randomized implementation and theoretical guarantees.

## Contribution

It proposes a novel robust PCA method that addresses both sparse and column-wise corruptions simultaneously, extending beyond existing algorithms.

## Key findings

- The algorithm effectively distinguishes inliers from outliers using sparse approximation.
- The approach is scalable with a randomized design.
- Theoretical analysis guarantees accurate sparse representation despite corruptions.

## Abstract

We study a data model in which the data matrix D can be expressed as D = L + S + C, where L is a low rank matrix, S an element-wise sparse matrix and C a matrix whose non-zero columns are outlying data points. To date, robust PCA algorithms have solely considered models with either S or C, but not both. As such, existing algorithms cannot account for simultaneous element-wise and column-wise corruptions. In this paper, a new robust PCA algorithm that is robust to simultaneous types of corruption is proposed. Our approach hinges on the sparse approximation of a sparsely corrupted column so that the sparse expansion of a column with respect to the other data points is used to distinguish a sparsely corrupted inlier column from an outlying data point. We also develop a randomized design which provides a scalable implementation of the proposed approach. The core idea of sparse approximation is analyzed analytically where we show that the underlying ell_1-norm minimization can obtain the representation of an inlier in presence of sparse corruptions.