A Converged Algorithm for Tikhonov Regularized Nonnegative Matrix Factorization with Automatic Regularization Parameters Determination

TL;DR

This paper introduces a converged algorithm for Tikhonov regularized nonnegative matrix factorization that automatically determines regularization parameters using the L-curve, addressing key issues of convergence and parameter selection.

Contribution

It proposes a novel converged algorithm for Tikhonov regularized NMF with an automatic regularization parameter selection mechanism based on the L-curve.

Findings

Algorithm guarantees convergence.

Automatic regularization parameter determination.

Improved suitability for linear inverse problems.

Abstract

We present a converged algorithm for Tikhonov regularized nonnegative matrix factorization (NMF). We specially choose this regularization because it is known that Tikhonov regularized least square (LS) is the more preferable form in solving linear inverse problems than the conventional LS. Because an NMF problem can be decomposed into LS subproblems, it can be expected that Tikhonov regularized NMF will be the more appropriate approach in solving NMF problems. The algorithm is derived using additive update rules which have been shown to have convergence guarantee. We equip the algorithm with a mechanism to automatically determine the regularization parameters based on the L-curve, a well-known concept in the inverse problems community, but is rather unknown in the NMF research. The introduction of this algorithm thus solves two inherent problems in Tikhonov regularized NMF algorithm research, i.e., convergence guarantee and regularization parameters determination.