Nonnegative Matrix Factorization with Toeplitz Penalty

TL;DR

This paper introduces a novel NMF algorithm incorporating Toeplitz matrix constraints, enhancing facial recognition performance compared to existing data-dependent constrained NMF methods.

Contribution

The paper proposes a new Toeplitz-constrained NMF algorithm that uses auxiliary constraints independent of data, improving facial recognition accuracy.

Findings

TNMF outperforms ZNMF in facial recognition tasks.

TNMF achieves higher accuracy on Cambridge ORL and Yale face databases.

The Toeplitz constraint enhances the interpretability and effectiveness of NMF.

Abstract

Nonnegative Matrix Factorization (NMF) is an unsupervised learning algorithm that produces a linear, parts-based approximation of a data matrix. NMF constructs a nonnegative low rank basis matrix and a nonnegative low rank matrix of weights which, when multiplied together, approximate the data matrix of interest using some cost function. The NMF algorithm can be modified to include auxiliary constraints which impose task-specific penalties or restrictions on the cost function of the matrix factorization. In this paper we propose a new NMF algorithm that makes use of non-data dependent auxiliary constraints which incorporate a Toeplitz matrix into the multiplicative updating of the basis and weight matrices. We compare the facial recognition performance of our new Toeplitz Nonnegative Matrix Factorization (TNMF) algorithm to the performance of the Zellner Nonnegative Matrix Factorization (ZNMF) algorithm which makes use of data-dependent auxiliary constraints. We also compare the facial recognition performance of the two aforementioned algorithms with the performance of several preexisting constrained NMF algorithms that have non-data-dependent penalties. The facial recognition performances are evaluated using the Cambridge ORL Database of Faces and the Yale Database of Faces.