Optimal smooth approximation for Quantile Matrix Factorization

<p dir="ltr">Matrix Factorization (MF) is essential to many estimation tasks. Most existing matrix factorization methods focus on least squares matrix factorization (LSMF), which aims to minimize a smooth L2 loss between observations and their dependent matrix measurement variables. In reality, however, L1 loss and check loss are widely used in regression to deal with outliers or observations contaminated by skewed or heavy-tailed noise.<br>Although under certain conditions, linear convergence to the global optimality can be established for matrix factorization under the L2 loss, there is a lack of provably efficient algorithms for solving matrix factorization under non-smooth losses. In this paper, we investigate Quantile Matrix Factorization (QMF), the counterpart of Quantile Regression in matrix estimation, that adopts a tunable check loss and introduces robustness to matrix estimation for 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 the matrix factorization setting. We then present an alternating minimization algorithm to solve the smooth NsQMF efficiently. We mathematically prove that solving the smoothed NsQMF is equivalent to solving the original non-smooth QMF problem and that our proposed algorithm achieves linear convergence to the global optimality of QMF. Numerical evaluations verify our theoretical findings and demonstrate that NsQMF significantly outperforms the commonly used LSMF and prior approximate smoothing heuristics for QMF under various noise distributions.</p>