# Probabilistic Simplex Component Analysis

arXiv: https://arxiv.org/abs/2103.10027 · HTML: https://mlsanity.com/paper/2103.10027/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 55 references that are in the map.
- First posted on arXiv 18 Mar 2021.

## Builds on (most cited first, 9 of 9)

- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications](https://mlsanity.com/paper/1803.01257/) (2018)
- [The Why and How of Nonnegative Matrix Factorization](https://mlsanity.com/paper/1401.5226/) (2014)
- [Hyperspectral Unmixing: Ground Truth Labeling, Datasets, Benchmark Performances and Survey](https://mlsanity.com/paper/1708.05125/) (2017)
- [Robust Volume Minimization-Based Matrix Factorization for Remote Sensing and Document Clustering](https://mlsanity.com/paper/1608.04290/) (2016)
- [Identifiability of the Simplex Volume Minimization Criterion for Blind Hyperspectral Unmixing: The No Pure-Pixel Case](https://mlsanity.com/paper/1406.5273/) (2014)
- [Crowdsourcing via Pairwise Co-occurrences: Identifiability and Algorithms](https://mlsanity.com/paper/1909.12325/) (2019)
- [Maximum Volume Inscribed Ellipsoid: A New Simplex-Structured Matrix Factorization Framework via Facet Enumeration and Convex Optimization](https://mlsanity.com/paper/1708.02883/) (2017)
- [Intersecting Faces: Non-negative Matrix Factorization With New Guarantees](https://mlsanity.com/paper/1507.02189/) (2015)

## Built on by (most cited first, 8 of 8)

- [Smoothed Separable Nonnegative Matrix Factorization](https://mlsanity.com/paper/2110.05528/) (2021)
- [On the Robustness of the Successive Projection Algorithm](https://mlsanity.com/paper/2411.16195/) (2024)
- [On Hyperspectral Unmixing](https://mlsanity.com/paper/2106.14177/) (2021)
- [A Bayesian Perspective for Determinant Minimization Based Robust Structured Matrix Factorizatio](https://mlsanity.com/paper/2302.08416/) (2023)
- [Hyperspectral Unmixing Under Endmember Variability: A Variational Inference Framework](https://mlsanity.com/paper/2407.14899/) (2024)
- [Robustness of Minimum-Volume Nonnegative Matrix Factorization under an Expanded Sufficiently Scattered Condition](https://mlsanity.com/paper/2511.04291/) (2025)
- [A Provably-Correct and Robust Convex Model for Smooth Separable NMF](https://mlsanity.com/paper/2511.07109/) (2025)
- [Scalable Minimum-Volume Simplex Estimation with Non-asymptotic Analysis](https://mlsanity.com/paper/2609.25576/) (2026)

## Abstract (arXiv)

This study presents PRISM, a probabilistic simplex component analysis approach to identifying the vertices of a data-circumscribing simplex from data. The problem has a rich variety of applications, the most notable being hyperspectral unmixing in remote sensing and non-negative matrix factorization in machine learning. PRISM uses a simple probabilistic model, namely, uniform simplex data distribution and additive Gaussian noise, and it carries out inference by maximum likelihood. The inference model is sound in the sense that the vertices are provably identifiable under some assumptions, and it suggests that PRISM can be effective in combating noise when the number of data points is large. PRISM has strong, but hidden, relationships with simplex volume minimization, a powerful geometric approach for the same problem. We study these fundamental aspects, and we also consider algorithmic schemes based on importance sampling and variational inference. In particular, the variational inference scheme is shown to resemble a matrix factorization problem with a special regularizer, which draws an interesting connection to the matrix factorization approach. Numerical results are provided to demonstrate the potential of PRISM.