On The Identifiability Of Transform Learning For Non-Negative Matrix Factorization

Non-negative matrix factorization with transform learning (TL-NMF) aims at estimating a short-time orthogonal transform that projects temporal data into a domain that is more amenable to NMF than off-the-shelf time-frequency transforms. In this work, we study the identifiability of TL-NMF under the Gaussian composite model.We prove that one can uniquely identify row-spaces of the orthogonal transform by optimizing the likelihood function of themodel. This result is illustrated on a toy source separation problem which demonstrates the ability of TL-NMF to learn a suitable orthogonal basis.