Chem-NMF: Multi-layer $α$-divergence Non-Negative Matrix Factorization for Cardiorespiratory Disease Clustering, with Improved Convergence Inspired by Chemical Catalysts and Rigorous Asymptotic Analysis

TL;DR

Chem-NMF tackles convergence challenges in multi-layer $\alpha$-divergence NMF by introducing a bounding factor inspired by chemical catalysts. It builds an energy-barrier framework with Boltzmann probability to characterize escape from local minima and proves non-increasing objectives via auxiliary functions, extended to a multi-layer setting. Empirically, Chem-NMF achieves meaningful clustering gains on both image and biomedical audio data, reporting mean improvements of $5.6\% \pm 2.7\%$ on biomedical signals and $11.1\% \pm 7.2\%$ on face images, with an overall ORL accuracy uplift of around $11\% \pm 7\%$ over baselines. The work provides a physically grounded, theoretically informed approach to stabilizing multi-layer NMF with practical impact for biomedical signal processing and image clustering.

Abstract

Non-Negative Matrix Factorization (NMF) is an unsupervised learning method offering low-rank representations across various domains such as audio processing, biomedical signal analysis, and image recognition. The incorporation of $α$-divergence in NMF formulations enhances flexibility in optimization, yet extending these methods to multi-layer architectures presents challenges in ensuring convergence. To address this, we introduce a novel approach inspired by the Boltzmann probability of the energy barriers in chemical reactions to theoretically perform convergence analysis. We introduce a novel method, called Chem-NMF, with a bounding factor which stabilizes convergence. To our knowledge, this is the first study to apply a physical chemistry perspective to rigorously analyze the convergence behaviour of the NMF algorithm. We start from mathematically proven asymptotic convergence results and then show how they apply to real data. Experimental results demonstrate that the proposed algorithm improves clustering accuracy by 5.6% $\pm$ 2.7% on biomedical signals and 11.1% $\pm$ 7.2% on face images (mean $\pm$ std).

Figures (10)

• Figure 1: Energy profile of a reaction progress: (a) reactants, intermediate, and products. The two transition states (TS1 and TS2) correspond to the energy maxima, with activation free energies $\Delta G^{\ddagger}_{1}$ and $\Delta G^{\ddagger}_{2}$ indicated by vertical arrows. The overall free energy change $\Delta G$ is shown between reactants and products. (b) catalyst effect on lowering the activation barrier.
• Figure 2: Overview of the clustering procedure. The input dataset $\mathbf{Y}$ is factorized into a feature basis $\mathbf{A}$ and activation maps $\mathbf{X}$ across multiple layers using NMF. The activation maps are clustered with $k$-means, and images are reconstructed using feature basis matrices.
• Figure 3: Example images from the ORL face and MNIST digit datasets under clean and noisy conditions. From top to bottom: clean images followed by Gaussian noise at 30, 20, 10, and 5 dB SNR levels.
• Figure 4: Time--frequency spectrograms from the HLS-CMDS dataset: (a) Lung sounds: CC: coarse crackles, FC: fine crackles, N: normal breathing, PR: pleural rub, R: rhonchi, W: wheeze; (b) Heart sounds: AF: atrial fibrillation, AVB: atrioventricular block, ESM: ejection systolic murmur, LDM: late diastolic murmur, LSM: late systolic murmur, MSM: mid-systolic murmur, NH: normal heart sound, S3: third heart sound, S4: fourth heart sound, T: tricuspid insufficiency.
• Figure 5: Effect of $\alpha$ value on the optimization landscape. Each subplot shows the trajectory for a specific $\alpha$: green points indicate the initialization, red points denote the final optimized solutions, and the black point marks the desired global minimum.

Theorems & Definitions (23)

• Theorem 3.1
• Definition 3.1: Auxiliary Function
• Lemma 3.1
• Theorem 3.2
• Definition 3.2: Energy Barrier
• Definition 3.3: Boltzmann Probability
• Lemma 3.2
• Theorem 3.3
• proof
• Corollary 3.1