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Optimal Smooth Approximation for Quantile Matrix Factorization

Liu, Peng, Liu, Yi, Zhu, Rui, Kong, Linglong, Jiang, Bei, Niu, Di (2023) Optimal Smooth Approximation for Quantile Matrix Factorization. In: Proceedings of the 2023 SIAM International Conference on Data Mining (SDM). . Society for Industrial and Applied Mathematics E-ISBN 978-1-61197-765-3. (doi:10.1137/1.9781611977653.ch67) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:98253)

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Abstract

Matrix Factorization (MF) is a critical technique in many applications. Most existing MF methods minimize the $L_2$ loss between observations and their dependent matrix measurement variables. Under certain conditions, linear convergence to global optimality can be established for $L_2$ loss, while $L_1$ loss and check loss are widely used to deal with data that are contaminated with outliers, there lacks of efficient and theoretically proven algorithms specifically designed for MF with non-smooth losses. In this paper, we study Quantile Matrix Factorization (QMF) that adopts a tunable check loss and can introduce robustness to matrix estimation under possibly skewed and heavy-tailed observations, which are prevalent in reality. To deal with the non-smooth loss, we propose Nesterov-smoothed QMF (NsQMF), extending Nesterov's optimal smooth approximation technique to MF. We then present an alternating minimization algorithm to solve NsQMF efficiently while handling non-convexity. We prove that solving the smoothed NsQMF is equivalent to solving the original non-smooth QMF problem and that our algorithm achieves linear convergence to the global optimality of QMF. Extensive evaluations verify our theoretical findings and demonstrate that NsQMF significantly outperforms commonly used Least Squares Matrix Factorization (LSMF) and prior rough smoothing techniques for QMF under various noise distributions.

Item Type: Conference or workshop item (Paper)
DOI/Identification number: 10.1137/1.9781611977653.ch67
Subjects: Q Science > QA Mathematics (inc Computing science) > QA276 Mathematical statistics
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Peng Liu
Date Deposited: 23 Nov 2022 09:36 UTC
Last Modified: 23 Jul 2023 17:20 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/98253 (The current URI for this page, for reference purposes)

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