# Identifiability of the Simplex Volume Minimization Criterion for Blind Hyperspectral Unmixing: The No Pure-Pixel Case

arXiv: https://arxiv.org/abs/1406.5273 · HTML: https://mlsanity.com/paper/1406.5273/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jun 2014.

## Built on by (most cited first, 26 of 26)

- [VB-LoRA: Extreme Parameter Efficient Fine-Tuning with Vector Banks](https://mlsanity.com/paper/2405.15179/) (2024)
- [Polytopic Matrix Factorization: Determinant Maximization Based Criterion and Identifiability](https://mlsanity.com/paper/2202.09638/) (2022)
- [Image Processing and Machine Learning for Hyperspectral Unmixing: An Overview and the HySUPP Python Package](https://mlsanity.com/paper/2308.09375/) (2023)
- [Probabilistic Simplex Component Analysis](https://mlsanity.com/paper/2103.10027/) (2021)
- [Fast and Structured Block-Term Tensor Decomposition For Hyperspectral Unmixing](https://mlsanity.com/paper/2205.03798/) (2022)
- [Biologically-Plausible Determinant Maximization Neural Networks for Blind Separation of Correlated Sources](https://mlsanity.com/paper/2209.12894/) (2022)
- [A Survey on Archetypal Analysis](https://mlsanity.com/paper/2504.12392/) (2025)
- [Smoothed Separable Nonnegative Matrix Factorization](https://mlsanity.com/paper/2110.05528/) (2021)
- [Bounded Simplex-Structured Matrix Factorization: Algorithms, Identifiability and Applications](https://mlsanity.com/paper/2209.12638/) (2022)
- [Deep Clustering with Incomplete Noisy Pairwise Annotations: A Geometric Regularization Approach](https://mlsanity.com/paper/2305.19391/) (2023)
- [Dual Simplex Volume Maximization for Simplex-Structured Matrix Factorization](https://mlsanity.com/paper/2403.20197/) (2024)
- [Correlative Information Maximization Based Biologically Plausible Neural Networks for Correlated Source Separation](https://mlsanity.com/paper/2210.04222/) (2022)
- [Checking the Sufficiently Scattered Condition using a Global Non-Convex Optimization Software](https://mlsanity.com/paper/2402.06019/) (2024)
- [On Hyperspectral Unmixing](https://mlsanity.com/paper/2106.14177/) (2021)
- [Identifiability of Nonnegative Tucker Decompositions -- Part I: Theory](https://mlsanity.com/paper/2505.12713/) (2025)
- [Towards the Identifiability in Noisy Label Learning: A Multinomial Mixture Modelling Approach](https://mlsanity.com/paper/2301.01405/) (2023)
- [Deep Nonnegative Matrix Factorization with Beta Divergences](https://mlsanity.com/paper/2309.08249/) (2023)
- [Towards Tuning-Free Minimum-Volume Nonnegative Matrix Factorization](https://mlsanity.com/paper/2309.13733/) (2023)
- [Block Majorization Minimization with Extrapolation and Application to β-NMF](https://mlsanity.com/paper/2401.06646/) (2024)
- [Quantum Information-Empowered Graph Neural Network for Hyperspectral Change Detection](https://mlsanity.com/paper/2411.07608/) (2024)
- [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)
- [Maximum-Volume Nonnegative Matrix Factorization](https://mlsanity.com/paper/2602.04795/) (2026)
- [Identification of NMF by choosing maximum-volume basis vectors](https://mlsanity.com/paper/2603.24227/) (2026)
- [Transformer-Driven Inverse Problem Transform for Fast Blind Hyperspectral Image Dehazing](https://mlsanity.com/paper/2501.01924/) (2025)
- [Robustness of Minimum-Volume Nonnegative Matrix Factorization under an Expanded Sufficiently Scattered Condition](https://mlsanity.com/paper/2511.04291/) (2025)
- [Normative Networks for Source Separation via Local Plasticity and Dendritic Computation](https://mlsanity.com/paper/2605.19965/) (2026)

## Abstract (arXiv)

In blind hyperspectral unmixing (HU), the pure-pixel assumption is well-known to be powerful in enabling simple and effective blind HU solutions. However, the pure-pixel assumption is not always satisfied in an exact sense, especially for scenarios where pixels are heavily mixed. In the no pure-pixel case, a good blind HU approach to consider is the minimum volume enclosing simplex (MVES). Empirical experience has suggested that MVES algorithms can perform well without pure pixels, although it was not totally clear why this is true from a theoretical viewpoint. This paper aims to address the latter issue. We develop an analysis framework wherein the perfect endmember identifiability of MVES is studied under the noiseless case. We prove that MVES is indeed robust against lack of pure pixels, as long as the pixels do not get too heavily mixed and too asymmetrically spread. The theoretical results are verified by numerical simulations.