# Robust Kronecker-Decomposable Component Analysis for Low-Rank Modeling

## TL;DR

This paper introduces a robust Kronecker-decomposable component analysis method that improves low-rank modeling of images, handling noise and outliers efficiently, and is applicable to background subtraction and image denoising.

## Contribution

It presents a novel Kronecker-decomposable approach combining sparse dictionary learning and PCP, with an efficient algorithm leveraging tensor factorization for robust image decomposition.

## Key findings

- Outperforms current methods in background subtraction.
- Effective in image denoising with robustness to corruptions.
- Reduces computational complexity compared to traditional SVD-based methods.

## Abstract

Dictionary learning and component analysis are part of one of the most well-studied and active research fields, at the intersection of signal and image processing, computer vision, and statistical machine learning. In dictionary learning, the current methods of choice are arguably K-SVD and its variants, which learn a dictionary (i.e., a decomposition) for sparse coding via Singular Value Decomposition. In robust component analysis, leading methods derive from Principal Component Pursuit (PCP), which recovers a low-rank matrix from sparse corruptions of unknown magnitude and support. However, K-SVD is sensitive to the presence of noise and outliers in the training set. Additionally, PCP does not provide a dictionary that respects the structure of the data (e.g., images), and requires expensive SVD computations when solved by convex relaxation. In this paper, we introduce a new robust decomposition of images by combining ideas from sparse dictionary learning and PCP. We propose a novel Kronecker-decomposable component analysis which is robust to gross corruption, can be used for low-rank modeling, and leverages separability to solve significantly smaller problems. We design an efficient learning algorithm by drawing links with a restricted form of tensor factorization. The effectiveness of the proposed approach is demonstrated on real-world applications, namely background subtraction and image denoising, by performing a thorough comparison with the current state of the art.