# Maximum-distance nonnegative matrix factorization for unmixing highly mixed grain-size distribution data: A generalization of AnalySize

arXiv: https://arxiv.org/abs/2608.22681 · HTML: https://mlsanity.com/paper/2608.22681/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 30 references that are in the map.
- First posted on arXiv 24 Aug 2026.

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

- [Projection onto the probability simplex: An efficient algorithm with a simple proof, and an application](https://mlsanity.com/paper/1309.1541/) (2013)
- [Nonnegative Matrix Factorization in Dimensionality Reduction: A Survey](https://mlsanity.com/paper/2405.03615/) (2024)
- [Unmixing highly mixed grain size distribution data via maximum volume constrained end member analysis](https://mlsanity.com/paper/2601.00154/) (2026)
- [Identification of NMF by choosing maximum-volume basis vectors](https://mlsanity.com/paper/2603.24227/) (2026)

## Abstract (arXiv)

Nonnegative matrix factorization (NMF) decomposes a nonnegative matrix into the product of two nonnegative matrices. This property makes NMF well suited for unmixing grain-size distribution data, which are inherently nonnegative and have row sums equal to one. Previous studies have shown that AnalySize, an NMF-based method, performs well on poorly mixed grain-size distribution data but struggles when the data is highly mixed, where no observed samples are close to the true end members. To overcome this limitation, we introduce a maximum-distance NMF that encourages the estimated end members to be as distinct as possible and develop a hierarchical alternating least squares algorithm for optimization. The proposed formulation can be regarded as a generalization of AnalySize, where AnalySize minimizes the distance among end members while the proposed method maximizes it. Experimental results demonstrate that the method effectively decomposes highly mixed grain-size distribution data.