{"ID":2921718,"CreatedAt":"2026-06-02T02:42:49.606572591Z","UpdatedAt":"2026-06-03T05:56:00.181519634Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.01216","arxiv_id":"2606.01216","title":"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 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.","short_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, w...","url_abs":"https://arxiv.org/abs/2606.01216","url_pdf":"https://arxiv.org/pdf/2606.01216v1","authors":"[\"Pratik Jawanpuria\",\"Ankish Chandresh\",\"Bamdev Mishra\"]","published":"2026-05-31T13:10:23Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.OC\"]","methods":"[]","has_code":false}