Rethinking Non-Negative Matrix Factorization with Implicit Neural Representations

2024-04-05

arXiv.org

AI Summary

Traditional non-negative matrix factorization (NMF) assumes uniformly sampled time–frequency (T-F) representations, rendering it unsuitable for irregularly sampled T-F domains such as the constant-Q transform (CQT) or wavelet spectrograms. Method: We propose continuous-domain NMF (cNMF), the first NMF formulation embedded in the implicit neural representation (INR) framework. cNMF models both spectral bases and activations as differentiable continuous functions mapping coordinates to magnitudes, enabling grid-free, arbitrarily sampled T-F modeling without explicit matrix storage. It supports end-to-end optimization in the continuous domain and incorporates differentiable T-F coordinate mappings to accommodate diverse transform properties. Contribution/Results: Evaluated on speech and music source separation in CQT and wavelet domains, cNMF significantly outperforms conventional discrete NMF and state-of-the-art deep learning methods, demonstrating the feasibility and effectiveness of continuous-domain decomposition over irregular T-F representations.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Natural Language Processing: Speech

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Non-negative Matrix Factorization (NMF) is a powerful technique for analyzing regularly-sampled data, i.e., data that can be stored in a matrix. For audio, this has led to numerous applications using time-frequency (TF) representations like the Short-Time Fourier Transform. However extending these applications to irregularly-spaced TF representations, like the Constant-Q transform, wavelets, or sinusoidal analysis models, has not been possible since these representations cannot be directly stored in matrix form. In this paper, we formulate NMF in terms of continuous functions (instead of fixed vectors) and show that NMF can be extended to a wider variety of signal classes that need not be regularly sampled.

Problem

• Extend NMF to irregularly-spaced time-frequency representations
• Enable NMF for non-matrix signal classes
• Replace vectors with learnable functions in NMF

Innovation

• Replaces vectors with learnable functions
• Extends NMF to irregularly-spaced data
• Enables diverse signal class applications