Exponentially Convergent Algorithms for Supervised Matrix Factorization
This addresses the problem of efficient and reliable training for SMF in high-dimensional data, such as in bioinformatics for cancer gene identification, though it is incremental in improving convergence guarantees.
The paper tackles the nonconvex optimization challenge in supervised matrix factorization (SMF) by proposing a novel framework that lifts SMF to a low-rank matrix estimation problem, resulting in an algorithm that provably converges exponentially fast to a global minimizer under mild assumptions.
Supervised matrix factorization (SMF) is a classical machine learning method that simultaneously seeks feature extraction and classification tasks, which are not necessarily a priori aligned objectives. Our goal is to use SMF to learn low-rank latent factors that offer interpretable, data-reconstructive, and class-discriminative features, addressing challenges posed by high-dimensional data. Training SMF model involves solving a nonconvex and possibly constrained optimization with at least three blocks of parameters. Known algorithms are either heuristic or provide weak convergence guarantees for special cases. In this paper, we provide a novel framework that 'lifts' SMF as a low-rank matrix estimation problem in a combined factor space and propose an efficient algorithm that provably converges exponentially fast to a global minimizer of the objective with arbitrary initialization under mild assumptions. Our framework applies to a wide range of SMF-type problems for multi-class classification with auxiliary features. To showcase an application, we demonstrate that our algorithm successfully identified well-known cancer-associated gene groups for various cancers.