Title: Riemannian Optimization for Hadamard Products of Low-Rank Matrices

Abstract: While the elementwise Hadamard product of two low-rank matrices offers a parameter-efficient framework for capturing multiplicative data structures, its effective modeling is complicated by inherent symmetries arising from coupled row and column scalings within the factor matrices. To address this, we cast the learning process as an optimization problem on a Riemannian quotient manifold, thereby exploiting the underlying geometric structure. We introduce a new block-diagonal Riemannian metric, obtained via the pullback of the Frobenius inner product, which demonstrates invariance under the entire symmetry group. Furthermore, we present a Riemannian gradient descent algorithm that incorporates a tuning-free Gauss–Newton step size and ensures linear computational scaling with respect to the number of observed entries in each iteration. Our empirical evaluations on both synthetic and real-world datasets confirm the effectiveness of this proposed Riemannian methodology.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC