# Orthogonal Nonnegative Matrix Factorization with Sparsity Constraints

arXiv: https://arxiv.org/abs/2210.02672 · HTML: https://mlsanity.com/paper/2210.02672/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 35 references that are in the map.
- First posted on arXiv 6 Oct 2022.

## Builds on (most cited first, 7 of 7)

- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Adaptive Gradient Descent without Descent](https://mlsanity.com/paper/1910.09529/) (2019)
- [Block Coordinate Descent for Sparse NMF](https://mlsanity.com/paper/1301.3527/) (2013)
- [Approximation Algorithms for Orthogonal Non-negative Matrix Factorization](https://mlsanity.com/paper/2103.01398/) (2021)
- [On the Persistence of Clustering Solutions and True Number of Clusters in a Dataset](https://mlsanity.com/paper/1811.00102/) (2018)
- [Sparse Separable Nonnegative Matrix Factorization](https://mlsanity.com/paper/2006.07553/) (2020)

## Built on by (most cited first, 1 of 1)

- [Autonomy-Aware Clustering: When Local Decisions Supersede Global Prescriptions](https://mlsanity.com/paper/2509.25775/) (2025)

## Abstract (arXiv)

This article presents a novel approach to solving the sparsity-constrained Orthogonal Nonnegative Matrix Factorization (SCONMF) problem, which requires decomposing a non-negative data matrix into the product of two lower-rank non-negative matrices, X=WH, where the mixing matrix H has orthogonal rows HH^T=I, while also satisfying an upper bound on the number of nonzero elements in each row. By reformulating SCONMF as a capacity-constrained facility-location problem (CCFLP), the proposed method naturally integrates non-negativity, orthogonality, and sparsity constraints. Specifically, our approach integrates control-barrier function (CBF) based framework used for dynamic optimal control design problems with maximum-entropy-principle-based framework used for facility location problems to enforce these constraints while ensuring robust factorization. Additionally, this work introduces a quantitative approach for determining the ``true" rank of W or H, equivalent to the number of ``true" features - a critical aspect in ONMF applications where the number of features is unknown. Simulations on various datasets demonstrate significantly improved factorizations with low reconstruction errors (as small as by 150 times) while strictly satisfying all constraints, outperforming existing methods that struggle with balancing accuracy and constraint adherence.