DeepAI AI Chat
Log In Sign Up

Maximum Volume Inscribed Ellipsoid: A New Simplex-Structured Matrix Factorization Framework via Facet Enumeration and Convex Optimization

by   Chia-Hsiang Lin, et al.
National Tsing Hua University
Virginia Polytechnic Institute and State University
The Chinese University of Hong Kong

Consider a structured matrix factorization scenario where one factor is modeled to have columns lying in the unit simplex. Such a simplex-structured matrix factorization (SSMF) problem has spurred much interest in key topics such as hyperspectral unmixing in remote sensing and topic discovery in machine learning. In this paper we develop a new theoretical framework for SSMF. The idea is to study a maximum volume ellipsoid inscribed in the convex hull of the data points, which has not been attempted in prior literature. We show a sufficient condition under which this maximum volume inscribed ellipsoid (MVIE) framework can guarantee exact recovery of the factors. The condition derived is much better than that of separable non-negative matrix factorization (or pure-pixel search) and is comparable to that of another powerful framework called minimum volume enclosing simplex. From the MVIE framework we also develop an algorithm that uses facet enumeration and convex optimization to achieve the aforementioned recovery result. Numerical results are presented to demonstrate the potential of this new theoretical SSMF framework.


page 1

page 2

page 3

page 4


Algorithms and Comparisons of Non-negative Matrix Factorization with Volume Regularization for Hyperspectral Unmixing

In this work, we consider nonnegative matrix factorization (NMF) with a ...

Robust Volume Minimization-Based Matrix Factorization for Remote Sensing and Document Clustering

This paper considers volume minimization (VolMin)-based structured matri...

Probabilistic Simplex Component Analysis

This study presents PRISM, a probabilistic simplex component analysis ap...

Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithms

In this paper, we provide novel algorithms with identifiability guarante...

Stochastic Matrix Factorization

This paper considers a restriction to non-negative matrix factorization ...

Polytopic Matrix Factorization: Determinant Maximization Based Criterion and Identifiability

We introduce Polytopic Matrix Factorization (PMF) as a novel data decomp...

Robust Vertex Enumeration for Convex Hulls in High Dimensions

Computation of the vertices of the convex hull of a set S of n points in...