Analysis of the rSVDdpd Algorithm: A Robust Singular Value Decomposition Method using Density Power Divergence

TL;DR

This paper investigates the theoretical properties and efficiency of the robust rSVDdpd algorithm, which improves singular value decomposition in the presence of outliers, especially in applications like video surveillance background modeling.

Contribution

It provides a detailed theoretical analysis of the rSVDdpd estimator's convergence, equivariance, and consistency, extending M-estimation theory for high-dimensional parameters.

Findings

rSVDdpd shows strong robustness against outliers.

Theoretical properties like convergence and consistency are established.

Simulation results demonstrate the method's efficiency.

Abstract

The traditional method of computing singular value decomposition (SVD) of a data matrix is based on a least squares principle, thus, is very sensitive to the presence of outliers. Hence the resulting inferences across different applications using the classical SVD are extremely degraded in the presence of data contamination (e.g., video surveillance background modelling tasks, etc.). A robust singular value decomposition method using the minimum density power divergence estimator (rSVDdpd) has been found to provide a satisfactory solution to this problem and works well in applications. For example, it provides a neat solution to the background modelling problem of video surveillance data in the presence of camera tampering. In this paper, we investigate the theoretical properties of the rSVDdpd estimator such as convergence, equivariance and consistency under reasonable assumptions. Since the dimension of the parameters, i.e., the number of singular values and the dimension of singular vectors can grow linearly with the size of the data, the usual M-estimation theory has to be suitably modified with concentration bounds to establish the asymptotic properties. We believe that we have been able to accomplish this satisfactorily in the present work. We also demonstrate the efficiency of rSVDdpd through extensive simulations.