Riemannian Optimization for Hadamard Products of Low-Rank Matrices
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918534318129152 |
|---|---|
| author | Jawanpuria, Pratik Chandresh, Ankish Mishra, Bamdev |
| author_facet | Jawanpuria, Pratik Chandresh, Ankish Mishra, Bamdev |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01216 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Riemannian Optimization for Hadamard Products of Low-Rank Matrices Jawanpuria, Pratik Chandresh, Ankish Mishra, Bamdev Machine Learning Optimization and Control 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. |
| title | Riemannian Optimization for Hadamard Products of Low-Rank Matrices |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2606.01216 |