Solving Principal Component Pursuit in Linear Time via $l_1$ Filtering

TL;DR

This paper introduces a novel $l_1$ filtering algorithm that efficiently solves the principal component pursuit problem in linear time, enabling large-scale robust PCA applications with high speed and parallelization.

Contribution

The paper presents the first linear-time algorithm for exactly solving nuclear norm minimization problems in robust PCA, significantly reducing computational complexity.

Findings

$l_1$ filtering achieves $O(r^2(m+n))$ complexity, much faster than traditional methods.

The algorithm is highly parallelizable, suitable for large-scale data.

Experiments demonstrate superior speed over existing algorithms.

Abstract

In the past decades, exactly recovering the intrinsic data structure from corrupted observations, which is known as robust principal component analysis (RPCA), has attracted tremendous interests and found many applications in computer vision. Recently, this problem has been formulated as recovering a low-rank component and a sparse component from the observed data matrix. It is proved that under some suitable conditions, this problem can be exactly solved by principal component pursuit (PCP), i.e., minimizing a combination of nuclear norm and $l_1$ norm. Most of the existing methods for solving PCP require singular value decompositions (SVD) of the data matrix, resulting in a high computational complexity, hence preventing the applications of RPCA to very large scale computer vision problems. In this paper, we propose a novel algorithm, called $l_1$ filtering, for \emph{exactly} solving PCP with an $O(r^2(m+n))$ complexity, where $m\times n$ is the size of data matrix and $r$ is the rank of the matrix to recover, which is supposed to be much smaller than $m$ and $n$. Moreover, $l_1$ filtering is \emph{highly parallelizable}. It is the first algorithm that can \emph{exactly} solve a nuclear norm minimization problem in \emph{linear time} (with respect to the data size). Experiments on both synthetic data and real applications testify to the great advantage of $l_1$ filtering in speed over state-of-the-art algorithms.