Towards Tuning-Free Minimum-Volume Nonnegative Matrix Factorization
This work addresses a tuning-free improvement for NMF users in data analysis, though it is incremental as it builds on existing minimum-volume NMF methods.
The paper tackles the problem of selecting a tuning parameter in minimum-volume nonnegative matrix factorization (NMF) that depends on unknown noise levels, proposing an alternative formulation inspired by the square-root lasso to make the optimal tuning parameter insensitive to noise, with empirical validation showing this insensitivity.
Nonnegative Matrix Factorization (NMF) is a versatile and powerful tool for discovering latent structures in data matrices, with many variations proposed in the literature. Recently, Leplat et al.\@ (2019) introduced a minimum-volume NMF for the identifiable recovery of rank-deficient matrices in the presence of noise. The performance of their formulation, however, requires the selection of a tuning parameter whose optimal value depends on the unknown noise level. In this work, we propose an alternative formulation of minimum-volume NMF inspired by the square-root lasso and its tuning-free properties. Our formulation also requires the selection of a tuning parameter, but its optimal value does not depend on the noise level. To fit our NMF model, we propose a majorization-minimization (MM) algorithm that comes with global convergence guarantees. We show empirically that the optimal choice of our tuning parameter is insensitive to the noise level in the data.