# Polytopic Matrix Factorization: Determinant Maximization Based Criterion and Identifiability

Source: [Polytopic Matrix Factorization: Determinant Maximization Based Criterion and Identifiability](<https://www.sota2.com/research/paper/68fa6bd51e209131cea3068f>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T23:57:05.246Z

Authors: Gokcan Tatli, Alper T. Erdogan

Published: 2022-02-19T16:49:24.000Z

## Abstract

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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

- [Sparse Source Separation on Sparse source realizations 30dB input SNR (test)](<https://www.sota2.com/research/sota/sparse-source-separation-on-sparse-source-realizations-30db-input-snr-test.md>): Sparse source separation; Sparse source realizations 30dB input SNR (test); SINR: 30.17.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2202.09638>)

[PDF](<https://arxiv.org/pdf/2202.09638.pdf>)