Sparse Hyperparametric Itakura-Saito Nonnegative Matrix Factorization via Bi-Level Optimization

The selection of penalty hyperparameters is a critical aspect in Nonnegative Matrix Factorization (NMF), since these values control the trade-off between reconstruction accuracy and adherence to desired constraints. In this work, we focus on an NMF problem involving the Itakura-Saito (IS) divergence, which is particularly effective for extracting low spectral density components from spectrograms of mixed signals, and benefits from the introduction of sparsity constraints. We propose a new algorithm called SHINBO, which introduces a bi-level optimization framework to automatically and adaptively tune the row-dependent penalty hyperparameters, enhancing the ability of IS-NMF to isolate sparse, periodic signals in noisy environments. Experimental results demonstrate that SHINBO achieves accurate spectral decompositions and demonstrates superior performance in both synthetic and real-world applications. In the latter case, SHINBO is particularly useful for noninvasive vibration-based fault detection in rolling bearings, where the desired signal components often reside in high-frequency subbands but are obscured by stronger, spectrally broader noise. By addressing the critical issue of hyperparameter selection, SHINBO improves the state-of-the-art in signal recovery for complex, noise-dominated environments.