# Non-negative Matrix Factorisation with Topological Regularisation

arXiv: https://arxiv.org/abs/2606.17531 · HTML: https://mlsanity.com/paper/2606.17531/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 65 references that are in the map.
- First posted on arXiv 16 Jun 2026.

## Builds on (most cited first, 18 of 18)

- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Graph Filtration Learning](https://mlsanity.com/paper/1905.10996/) (2019)
- [A Topological Regularizer for Classifiers via Persistent Homology](https://mlsanity.com/paper/1806.10714/) (2018)
- [A Framework for Differential Calculus on Persistence Barcodes](https://mlsanity.com/paper/1910.00960/) (2019)
- [Cubical Ripser: Software for computing persistent homology of image and volume data](https://mlsanity.com/paper/2005.12692/) (2020)
- [Topology-Aware Surface Reconstruction for Point Clouds](https://mlsanity.com/paper/1811.12543/) (2018)
- [Contact patterns among high school students](https://mlsanity.com/paper/1409.5318/) (2014)
- [Continuation of Point Clouds via Persistence Diagrams](https://mlsanity.com/paper/1506.03147/) (2015)
- [NIMFA: A Python Library for Nonnegative Matrix Factorization](https://mlsanity.com/paper/1808.01743/) (2018)
- [Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Clouds](https://mlsanity.com/paper/2307.09259/) (2023)
- [The Vietoris-Rips complexes of a circle](https://mlsanity.com/paper/1503.03669/) (2015)
- [Analyzing Single Cell RNA Sequencing with Topological Nonnegative Matrix Factorization](https://mlsanity.com/paper/2310.15744/) (2023)
- [Supervised low-rank semi-nonnegative matrix factorization with frequency regularization for forecasting spatio-temporal data](https://mlsanity.com/paper/2311.08636/) (2023)

## Abstract (arXiv)

Non-negative matrix factorisation (NMF) learns additive representations from data, but non-negativity alone does not ensure interpretable basis functions. We introduce Top-NMF, which guides basis learning through general topological preferences without prescribing the detailed form of the components. Observations and basis vectors are treated as non-negative functions on structured domains. Discrete topological conditions are difficult to optimise directly. Our guiding viewpoint is that persistent homology provides a stable, continuous relaxation of ordinary homological invariants: it tracks connected components and loops across thresholds, allowing structural preferences to be expressed through continuous scores suitable for optimisation. We construct such scores for connected image parts, clique-like graph structure, and periodic time-series components, and incorporate them into a common NMF objective. For graph data, we derive an exact maximum-weight-spanning-tree formula and characterise all local minimisers of the score, showing that their positive-weight edges form disjoint cliques. We establish the regularity of the scores, describe their derivatives, and analyse projected first-order optimisation. Numerical studies on synthetic and real data demonstrate how these priors guide basis learning and examine the balance between structural agreement, atom recovery, and reconstruction accuracy.