Expectile Matrix Factorization for Skewed Data Analysis

TL;DR

This paper introduces expectile matrix factorization, an approach that uses asymmetric least squares to better analyze skewed and extreme data, outperforming traditional least squares methods in accuracy.

Contribution

The paper develops a novel expectile matrix factorization method incorporating asymmetric least squares and provides an efficient algorithm with proven convergence and exact recovery guarantees.

Findings

Achieves lower recovery errors on skewed synthetic data.

Outperforms traditional least squares matrix factorization on real web response time data.

Provides theoretical convergence and recovery guarantees.

Abstract

Matrix factorization is a popular approach to solving matrix estimation problems based on partial observations. Existing matrix factorization is based on least squares and aims to yield a low-rank matrix to interpret the conditional sample means given the observations. However, in many real applications with skewed and extreme data, least squares cannot explain their central tendency or tail distributions, yielding undesired estimates. In this paper, we propose \emph{expectile matrix factorization} by introducing asymmetric least squares, a key concept in expectile regression analysis, into the matrix factorization framework. We propose an efficient algorithm to solve the new problem based on alternating minimization and quadratic programming. We prove that our algorithm converges to a global optimum and exactly recovers the true underlying low-rank matrices when noise is zero. For synthetic data with skewed noise and a real-world dataset containing web service response times, the proposed scheme achieves lower recovery errors than the existing matrix factorization method based on least squares in a wide range of settings.