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

TL;DR

This work tackles penalty hyperparameter tuning in Nonnegative Matrix Factorization under the Itakura-Saito divergence by introducing SHINBO, a bi-level optimization framework that learns row-wise penalty parameters for the activations. By formulating the inner problem with an IS-divergence and a row-diversity penalty, and solving it via a dynamical-system approximation plus forward-mode hypergradients, SHINBO automatically adjusts penalties to emphasize sparse, periodic components. Empirical results on synthetic data and real bearing vibration signals show SHINBO achieving higher spectral-identification quality (SIR) and spectral-structure metrics (ENVSI) while producing more interpretable sparse activations, outperforming standard MU and fixed-penalty variants. The approach reduces the need for manual hyperparameter tuning and enhances signal recovery in noise-dominated spectrograms, with practical impact in fault detection and spectral decomposition tasks across noisy domains.

Abstract

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.

Figures (7)

• Figure 1: The test rig used in the experiment GABOR2023110430.
• Figure 2: Recorded vibration signal and its spectrogram.
• Figure 3: Initial factors generated using $(1.5| \mathcal{N}|+0.5)/2$. Left: $\bm{W}_0\bm{H}_0$. Right: each rank-1 component in $\bm{W}_0\bm{H}_0$.
• Figure 4: Behavior of the Response function (outer problem) for the synthetic dataset.
• Figure 5: Results of sparsity and SIR on the factor matrices on the synthetic dataset. The left two columns are the sparsity of $\bm{W}$ (left) and $\bm{H}$ (right) of the four algorithms: from left to right are MU with $\lambda=0$, MU with $\lambda=0.1$, MU with $\lambda=0.5$, and SHINBO. The right two columns are SIR of $\bm{W}$ (left) and $\bm{H}$ (right) of the four algorithms: from the left to right are MU with $\lambda=0$, MU with $\lambda=0.1$, MU with $\lambda=0.5$, and SHINBO.

Theorems & Definitions (2)

• Remark 1
• Remark 2