# Adaptive Quantile Low-Rank Matrix Factorization

## TL;DR

This paper introduces AQ-LRMF, a novel low-rank matrix factorization method that models asymmetric noise with a mixture of Laplace distributions, improving robustness and accuracy in real-world noisy data.

## Contribution

The paper proposes a new LRMF model using asymmetric Laplace mixture noise modeling and an EM-based algorithm for parameter estimation, enhancing noise approximation capabilities.

## Key findings

- AQ-LRMF outperforms state-of-the-art methods on synthetic and real datasets.
- It effectively captures local structural information in images.
- The model handles both symmetric and skewed noise well.

## Abstract

Low-rank matrix factorization (LRMF) has received much popularity owing to its successful applications in both computer vision and data mining. By assuming noise to come from a Gaussian, Laplace or mixture of Gaussian distributions, significant efforts have been made on optimizing the (weighted) $L_1$ or $L_2$-norm loss between an observed matrix and its bilinear factorization. However, the type of noise distribution is generally unknown in real applications and inappropriate assumptions will inevitably deteriorate the behavior of LRMF. On the other hand, real data are often corrupted by skew rather than symmetric noise. To tackle this problem, this paper presents a novel LRMF model called AQ-LRMF by modeling noise with a mixture of asymmetric Laplace distributions. An efficient algorithm based on the expectation-maximization (EM) algorithm is also offered to estimate the parameters involved in AQ-LRMF. The AQ-LRMF model possesses the advantage that it can approximate noise well no matter whether the real noise is symmetric or skew. The core idea of AQ-LRMF lies in solving a weighted $L_1$ problem with weights being learned from data. The experiments conducted on synthetic and real datasets show that AQ-LRMF outperforms several state-of-the-art techniques. Furthermore, AQ-LRMF also has the superiority over the other algorithms in terms of capturing local structural information contained in real images.