# Polytopic Matrix Factorization: Determinant Maximization Based Criterion and Identifiability

arXiv: https://arxiv.org/abs/2202.09638 · HTML: https://mlsanity.com/paper/2202.09638/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 46 references that are in the map.
- First posted on arXiv 19 Feb 2022.

## Builds on (most cited first, 7 of 7)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications](https://mlsanity.com/paper/1803.01257/) (2018)
- [On Identifiability of Nonnegative Matrix Factorization](https://mlsanity.com/paper/1709.00614/) (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)
- [Maximum Volume Inscribed Ellipsoid: A New Simplex-Structured Matrix Factorization Framework via Facet Enumeration and Convex Optimization](https://mlsanity.com/paper/1708.02883/) (2017)
- [Blind Audio Source Separation with Minimum-Volume Beta-Divergence NMF](https://mlsanity.com/paper/1907.02404/) (2019)

## Built on by (most cited first, 14 of 14)

- [Self-Supervised Learning with an Information Maximization Criterion](https://mlsanity.com/paper/2209.07999/) (2022)
- [Biologically-Plausible Determinant Maximization Neural Networks for Blind Separation of Correlated Sources](https://mlsanity.com/paper/2209.12894/) (2022)
- [Dual Simplex Volume Maximization for Simplex-Structured Matrix Factorization](https://mlsanity.com/paper/2403.20197/) (2024)
- [Range, not Independence, Drives Modularity in Biologically Inspired Representations](https://mlsanity.com/paper/2410.06232/) (2024)
- [Diverse Influence Component Analysis: A Geometric Approach to Nonlinear Mixture Identifiability](https://mlsanity.com/paper/2510.17040/) (2025)
- [Correlative Information Maximization Based Biologically Plausible Neural Networks for Correlated Source Separation](https://mlsanity.com/paper/2210.04222/) (2022)
- [Correlative Information Maximization: A Biologically Plausible Approach to Supervised Deep Neural Networks without Weight Symmetry](https://mlsanity.com/paper/2306.04810/) (2023)
- [Checking the Sufficiently Scattered Condition using a Global Non-Convex Optimization Software](https://mlsanity.com/paper/2402.06019/) (2024)
- [Identifiable Shared Component Analysis of Unpaired Multimodal Mixtures](https://mlsanity.com/paper/2409.19422/) (2024)
- [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)
- [A Bayesian Perspective for Determinant Minimization Based Robust Structured Matrix Factorizatio](https://mlsanity.com/paper/2302.08416/) (2023)
- [Normative Networks for Source Separation via Local Plasticity and Dendritic Computation](https://mlsanity.com/paper/2605.19965/) (2026)
- [How Optimality Structures Sparse Dictionaries: Theory for Interpreting SAE Representations](https://mlsanity.com/paper/2606.02385/) (2026)

## Abstract (arXiv)

We introduce Polytopic Matrix Factorization (PMF) as a novel data decomposition approach. In this new framework, we model input data as unknown linear transformations of some latent vectors drawn from a polytope. In this sense, the article considers a semi-structured data model, in which the input matrix is modeled as the product of a full column rank matrix and a matrix containing samples from a polytope as its column vectors. The choice of polytope reflects the presumed features of the latent components and their mutual relationships. As the factorization criterion, we propose the determinant maximization (Det-Max) for the sample autocorrelation matrix of the latent vectors. We introduce a sufficient condition for identifiability, which requires that the convex hull of the latent vectors contains the maximum volume inscribed ellipsoid of the polytope with a particular tightness constraint. Based on the Det-Max criterion and the proposed identifiability condition, we show that all polytopes that satisfy a particular symmetry restriction qualify for the PMF framework. Having infinitely many polytope choices provides a form of flexibility in characterizing latent vectors. In particular, it is possible to define latent vectors with heterogeneous features, enabling the assignment of attributes such as nonnegativity and sparsity at the subvector level. The article offers examples illustrating the connection between polytope choices and the corresponding feature representations.