# Non-negative Matrix Factorization via Archetypal Analysis

arXiv: https://arxiv.org/abs/1705.02994 · HTML: https://mlsanity.com/paper/1705.02994/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 May 2017.

## Built on by (most cited first, 11 of 11)

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- [SpeedCP: Fast Kernel-based Conditional Conformal Prediction](https://mlsanity.com/paper/2509.24100/) (2025)
- [Nonnegative matrix factorizations and related compositional models: Equivalence, identifiability, and an application on the grain-size analysis of sediments](https://mlsanity.com/paper/2512.22282/) (2025)
- [Time series forecasting from partial observations via Non-negative Matrix Factorization](https://mlsanity.com/paper/2102.05314/) (2021)
- [Interpretable factorization of clinical questionnaires to identify latent factors of psychopathology](https://mlsanity.com/paper/2312.07762/) (2023)
- [BenchMake: Turn any scientific data set into a reproducible benchmark](https://mlsanity.com/paper/2506.23419/) (2025)
- [Semi-supervised Vertex Hunting, with Applications in Network and Text Analysis](https://mlsanity.com/paper/2510.22526/) (2025)

## Abstract (arXiv)

Given a collection of data points, non-negative matrix factorization (NMF) suggests to express them as convex combinations of a small set of `archetypes' with non-negative entries. This decomposition is unique only if the true archetypes are non-negative and sufficiently sparse (or the weights are sufficiently sparse), a regime that is captured by the separability condition and its generalizations. In this paper, we study an approach to NMF that can be traced back to the work of Cutler and Breiman (1994) and does not require the data to be separable, while providing a generally unique decomposition. We optimize the trade-off between two objectives: we minimize the distance of the data points from the convex envelope of the archetypes (which can be interpreted as an empirical risk), while minimizing the distance of the archetypes from the convex envelope of the data (which can be interpreted as a data-dependent regularization). The archetypal analysis method of (Cutler, Breiman, 1994) is recovered as the limiting case in which the last term is given infinite weight. We introduce a `uniqueness condition' on the data which is necessary for exactly recovering the archetypes from noiseless data. We prove that, under uniqueness (plus additional regularity conditions on the geometry of the archetypes), our estimator is robust. While our approach requires solving a non-convex optimization problem, we find that standard optimization methods succeed in finding good solutions both for real and synthetic data.