Leveraging Joint-Diagonalization in Transform-Learning NMF
This is an incremental improvement for researchers in matrix factorization and signal processing, addressing computational efficiency and robustness in data representation learning.
The paper tackled the problem of learning data representations for non-negative matrix factorization (NMF) by relating transform-learning NMF (TL-NMF) to joint-diagonalization (JD), showing that JD+NMF works with many data realizations but TL-NMF's low-rank constraint is essential for limited data.
Non-negative matrix factorization with transform learning (TL-NMF) is a recent idea that aims at learning data representations suited to NMF. In this work, we relate TL-NMF to the classical matrix joint-diagonalization (JD) problem. We show that, when the number of data realizations is sufficiently large, TL-NMF can be replaced by a two-step approach -- termed as JD+NMF -- that estimates the transform through JD, prior to NMF computation. In contrast, we found that when the number of data realizations is limited, not only is JD+NMF no longer equivalent to TL-NMF, but the inherent low-rank constraint of TL-NMF turns out to be an essential ingredient to learn meaningful transforms for NMF.