# Optimization on the Oblique Manifold for Sparse Simplex Constraints via Multiplicative Updates

arXiv: https://arxiv.org/abs/2503.24075 · HTML: https://mlsanity.com/paper/2503.24075/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 26 references that are in the map.
- First posted on arXiv 31 Mar 2025.

## Builds on (most cited first, 4 of 4)

- [Algorithms and Comparisons of Non-negative Matrix Factorization with Volume Regularization for Hyperspectral Unmixing](https://mlsanity.com/paper/1903.04362/) (2019)
- [From the simplex to the sphere: Faster constrained optimization using the Hadamard parametrization](https://mlsanity.com/paper/2112.05273/) (2021)
- [Fast Projection onto the Capped Simplex with Applications to Sparse Regression in Bioinformatics](https://mlsanity.com/paper/2110.08471/) (2021)
- [Theoretical aspects in penalty hyperparameters optimization](https://mlsanity.com/paper/2301.13542/) (2023)

## Abstract (arXiv)

Low-rank optimization problems with sparse simplex constraints involve variables that must satisfy nonnegativity, sparsity, and sum-to-1 conditions, making their optimization particularly challenging due to the interplay between low-rank structures and constraints. These problems arise in various applications, including machine learning, signal processing, environmental fields, and computational biology. In this work, we propose a novel manifold optimization approach to efficiently tackle these problems. Our method leverages the geometry of oblique manifolds to reformulate the problem and introduces a new Riemannian optimization method based on Riemannian gradient descent that strictly maintains the simplex constraints. By exploiting the underlying manifold structure, our approach improves optimization efficiency. Experiments on synthetic and real datasets demonstrate the effectiveness of the proposed method compared to standard Euclidean and Riemannian methods, paving the way for broader applications.