Maximum-distance nonnegative matrix factorization for unmixing highly mixed grain-size distribution data

Qianqian Qi; Zhongming Chen; Peter G. M. van der Heijden

cs.LGAug 24, 2026

Maximum-distance nonnegative matrix factorization for unmixing highly mixed grain-size distribution data

Authors: Qianqian Qi, Zhongming Chen, Peter G. M. van der Heijden

Organizations: Hangzhou Dianzi University, China · Utrecht University, the Netherlands · University of Southampton, UK

Abstract

Unmixing grain-size distribution data into several end-members can help reveal sediment provenance, transport, and sedimentation processes. Traditional end-member unmixing methods seek end-members closely to specimens, which is suitable when data is poorly mixed. However, these methods can be limited for highly mixed data when the observed specimens are not near to the underlying end-members. To overcome this limitation, we introduce a maximum-distance nonnegative matrix factorization model that encourages the estimated end-members to be as distinct as possible. We adopt a hierarchical alternating least squares algorithm for optimization and analyze the convergence properties. We evaluate its performance on simulated grain-size distribution data and compare it with existing method. The results illustrate the potential of the proposed approach for end-member unmixing in highly mixed data.

Figures & tables

Panel of Figure 1 — Figure 1: True end members

Panel of Figure 2 — Figure 2: Highly mixed two-end-member dataset: $\alpha=5$ , $\lambda=1$ (a and b) AnalySize; (c and d) MAD-NMF.

Panel of Figure 3 — Figure 3: Noisy highly mixed two-end-member dataset: $\alpha=5$ , $\lambda=1$ (a and b) AnalySize; (c and d) MAD-NMF.

Methods	AnalySize	MAD-NMF
Highly mixed two-end-member dataset
MAEM	10.2711	0.0245
MAAB	6.8464	0.0303
Noisy highly mixed two-end-member dataset
MAEM	10.2701	0.1186
MAAB	6.8387	0.0609

Table 1: MAEM and MAAB for (noisy) highly mixed two-end-member dataset

Methods	AnalySize	MAD-NMF
Highly mixed two-end-member dataset	0.0269	0.0549
Noisy highly mixed two-end-member dataset	0.0269	0.0548

Table 2: The

||\bm{C}_{K\times K}\bm{S}||_{F}^{2}

for (noisy) highly mixed two-end-member dataset

Panel of Figure 4 — Figure 4: Highly mixed coversand dataset: $\alpha=100$ , $\lambda=2.6$ (a and b) AnalySize; (c and d) MAD-NMF.

Panel of Figure 5 — Figure 5: Noisy highly mixed coversand dataset: $\alpha=100$ , $\lambda=2.6$ (a and b) AnalySize; (c and d) MAD-NMF.

Methods	AnalySize	MAD-NMF
Highly mixed coversand dataset
MAEM	15.6167	1.5787
MAAB	14.9915	1.4541
Noisy highly mixed coversand dataset
MAEM	15.5002	2.0862
MAAB	14.9172	3.1873

Table 3: MAEM and MAAB for (noisy) highly mixed coversand dataset

Methods	AnalySize	MAD-NMF
Highly mixed coversand dataset	0.0465	0.1801
Noisy highly mixed coversand dataset	0.0474	0.1907

Table 4: The

||\bm{C}_{K\times K}\bm{S}||_{F}^{2}

for (noisy) highly mixed coversand dataset

Non-Negative Matrix Factorization

Non-negative matrix factorization (NMF) decomposes nonnegative data into additive, interpretable latent components for analysis and discovery.

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Nonnegative matrix factorizations and related compositional models: Equivalence, identifiability, and an application on the grain-size analysis of sediments

Qianqian Qi, Peter G. M. van der Heijden, Maarten A. Prins

Hangzhou Dianzi University, China · Utrecht University, the Netherlands · University of Southampton, UK+1

Latent Variable Models Parameter Identifiability Non-Negative Matrix Factorization

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Across fields such as machine learning, social science, and geology, considerable attention has been given to models that factorize a nonnegative matrix into the product of two or three matrices, subject to nonnegative or row-sum-to-1 constraints. Although these models are to a large extent similar or even equivalent, they are presented under different names, and their similarity is not well known. This paper highlights similarities among five models, latent budget analysis (LBA) and latent class analysis (LCA) from social science, end-member analysis (EMA) from geology, probabilistic latent semantic analysis (PLSA) and nonnegative matrix factorization (NMF) from machine learning. We focus on the identifiability of these models. We prove that the solution of LBA, EMA, LCA, PLSA is unique if and only if the solution of NMF is unique. Consequently, existing uniqueness theorems for NMF directly apply to LBA, EMA, LCA, PLSA, and vice versa. We also provide a brief review of algorithms for the estimation of these models. We illustrate NMF on a sedimentary grain-size distribution dataset from sedimentary geology, and end the paper with a discussion of closely related model: archetypal analysis.

Abstract

Non-Negative Matrix Factorization

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