Dimension reduction of compositional data through mixtures of latent source compositions
Abstract
Understanding the distribution of mineral compositions is an important challenge in geology. This motivates new variations on dimension reduction for compositional data, lying on a simplex, that approximate observations with lower-dimensional subsimplices. This representation interprets compositions as convex combinations of fewer number of latent source compositions, providing unique insight into the underlying data-generating processes. We develop this approach within the framework of backwards principal component analysis, which constructs nested approximating structures by successive dimension reduction by rank one. We introduce a class of subsimplices, termed proper subsimplices, and provide a formal definition together with a systematic study of their geometric properties. Proper subsimplices admit effective parametrization via hyperplanes, while providing a more flexible approximation than subsimplices obtained by amalgamation. We characterize simple conditions under which hyperplanes generate proper subsimplices. Based on these conditions, we propose both an exact algorithm and a computationally efficient approximate algorithm for estimating proper subsimplex representations. The usefulness of this method is demonstrated on two geochemical datasets where it gives improved insights compared to the conventional PCA approach. Main ideas are further illustrated in a toy example and a simulation study.
Related
- Dimension-free comparison inequalities between f-divergences for Gaussian location mixtures
- Convex Order Beyond Dimension One: Projection Tests, Counterexamples and Gaussian Mixtures
- Dimension reduction for characterizing sexual dimorphism in biomechanics of the temporomandibular joint
- Robust conditional dimension reduction for dissimilarity data
- Algebraic Dimension and Reduction of LVMB manifolds
BibTeX
@misc{lee2026dimension,
title = {{Dimension reduction of compositional data through mixtures of latent source compositions}},
author = {Hyeon Lee and J. S. Marron and Eric Grunsky and JAnice L. Scealy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21173},
eprint = {2610.21173},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21173},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Dimension reduction of compositional data through mixtures of latent source compositions
AU - Hyeon Lee
AU - J. S. Marron
AU - Eric Grunsky
AU - JAnice L. Scealy
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21173
UR - https://arxiv.org/abs/2610.21173
AB - Understanding the distribution of mineral compositions is an important challenge in geology. This motivates new variations on dimension reduction for compositional data, lying on a simplex, that approximate observations with lower-dimensional subsimplices. This representation interprets compositions as convex combinations of fewer number of latent source compositions, providing unique insight into the underlying data-generating processes. We develop this approach within the framework of backwards principal component analysis, which constructs nested approximating structures by successive dimension reduction by rank one. We introduce a class of subsimplices, termed proper subsimplices, and provide a formal definition together with a systematic study of their geometric properties. Proper subsimplices admit effective parametrization via hyperplanes, while providing a more flexible approximation than subsimplices obtained by amalgamation. We characterize simple conditions under which hyperplanes generate proper subsimplices. Based on these conditions, we propose both an exact algorithm and a computationally efficient approximate algorithm for estimating proper subsimplex representations. The usefulness of this method is demonstrated on two geochemical datasets where it gives improved insights compared to the conventional PCA approach. Main ideas are further illustrated in a toy example and a simulation study.
ER -