Beyond Low Rank + Sparse: Multi-scale Low Rank Matrix Decomposition

TL;DR

This paper introduces a multi-scale low rank matrix decomposition method that captures local correlations at multiple scales, improving interpretability and effectiveness in various applications like image processing and medical imaging.

Contribution

It generalizes low rank + sparse decomposition to multi-scale low rank modeling with a convex approach, providing theoretical guarantees and practical guidance.

Findings

Effective in illumination normalization for face images

Improves motion separation in surveillance videos

Enhances multi-scale modeling in medical imaging

Abstract

We present a natural generalization of the recent low rank + sparse matrix decomposition and consider the decomposition of matrices into components of multiple scales. Such decomposition is well motivated in practice as data matrices often exhibit local correlations in multiple scales. Concretely, we propose a multi-scale low rank modeling that represents a data matrix as a sum of block-wise low rank matrices with increasing scales of block sizes. We then consider the inverse problem of decomposing the data matrix into its multi-scale low rank components and approach the problem via a convex formulation. Theoretically, we show that under various incoherence conditions, the convex program recovers the multi-scale low rank components \revised{either exactly or approximately}. Practically, we provide guidance on selecting the regularization parameters and incorporate cycle spinning to reduce blocking artifacts. Experimentally, we show that the multi-scale low rank decomposition provides a more intuitive decomposition than conventional low rank methods and demonstrate its effectiveness in four applications, including illumination normalization for face images, motion separation for surveillance videos, multi-scale modeling of the dynamic contrast enhanced magnetic resonance imaging and collaborative filtering exploiting age information.