Riemannian Optimization for Hadamard Products of Low-Rank Matrices

• URL: http://arxiv.org/abs/2606.01216v1
• Date: Sun, 31 May 2026 13:10:23 GMT
• Title: Riemannian Optimization for Hadamard Products of Low-Rank Matrices
• Abstract summary: An 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.<n>In order to leverage the geometry of the space, we formulate the learning of such matrices as optimization on a Riemannian quotient manifold.<n>We develop a gradient-free Gauss--Newton descent algorithm that uses a tuning-free Gauss--Newton step size and scales linearly in the number of observed entries per iteration.
• Score: 8.444687277893827
• License: http://creativecommons.org/licenses/by/4.0/
• 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 the full symmetry group. 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. Experiments on real and synthetic datasets illustrate the efficacy of our proposed Riemannian approach.

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