Riemannian Optimization for Hadamard Products of Low-Rank Matrices

Abstract:The elementwise Hadamard product of two low-rank matrices provides a parameter-efficient model for data with multiplicative structure, but its modeling is challenging due to the presence of additional symmetries under coupled row/column scalings between the two factors. In order to leverage the geometry of the space, we formulate the learning of such matrices as optimization on a Riemannian quotient manifold. We propose a novel block-diagonal Riemannian metric derived from the pullback of the Frobenius inner product. The metric is shown to be invariant under these symmetries. We develop a Riemannian gradient descent algorithm that uses a tuning-free Gauss--Newton step size and scales linearly in the number of observed entries per iteration. The versatile framework of Riemannian quotient optimization enables both first-order and second-order Riemannian methods, the latter through a closed-form connection and the Riemannian Hessian. Experiments on real and synthetic datasets illustrate the efficacy of our proposed Riemannian approach.

Machine Learning (cs.LG); Optimization and Control (math.OC)

arXiv:2606.01216 [cs.LG]

(or arXiv:2606.01216v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2606.01216 Focus to learn more arXiv-issued DOI via DataCite

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