Rethinking Non-Negative Matrix Factorization with Implicit Neural Representations

TL;DR

• NMF traditionally analyzes regularly sampled data stored in matrices, with audio research often using time-frequency representations like the Short-Time Fourier Transform (STFT).
• The authors propose reformulating NMF in terms of learnable functions (instead of fixed vectors), enabling extension to irregularly sampled signal classes such as Constant-Q transforms, wavelets, or sinusoidal models.
• The work highlights acceptance at WASPAA 2025 and references NeurIPS in the publication context, underscoring its cross-conference relevance.
• This approach broadens the applicability of NMF, potentially impacting audio analysis pipelines that rely on nonstandard TF representations.
• The paper is attributed to authors including Krishna Subramani, Paris Smaragdis, Takuya Higuchi, and Mehrez Souden. Apple Machine Learning Research article

Context and background

Non-Negative Matrix Factorization (NMF) has become a foundational tool for analyzing data that can be organized into a matrix, particularly in audio where time-frequency (TF) representations such as the Short-Time Fourier Transform (STFT) are widely used. NMF decomposes data into parts-based factors that are non-negative, a property that aligns well with many real-world audio phenomena like spectral magnitudes and energy distributions. However, a notable limitation arises when researchers want to work with irregularly spaced TF representations. Representations such as the Constant-Q transform, wavelets, or sinusoidal analysis models do not naturally fit into a matrix form that NMF can directly factorize. This mismatch constrains the ability to apply standard NMF techniques to a broad class of signals that are valuable in audio analysis. Apple’s research description emphasizes that the core barrier is the incompatibility between irregular sampling grids and matrix-based factorization. When TF representations cannot be stored as a matrix, the traditional NMF framework cannot be applied in a straightforward manner. As a result, researchers have faced a limitation in extending NMF to certain signals or in leveraging modern neural-powered representations within a matrix-based factorization paradigm. The paper under discussion positions itself within this gap by rethinking the mathematical formulation of NMF to accommodate learnable, function-based factors rather than fixed, vector-based factors. This shift lays the groundwork for applying NMF concepts to a wider range of signal classes that do not naturally conform to regular sampling. The work is associated with dates around December 9, 2024 and spans research areas including Methods and Algorithms as well as Speech and Natural Language Processing, with a conference history linked to NeurIPS and WASPAA in 2025. These cross-domain connections reflect a broader interest in bringing factorization techniques into learnable-function regimes that align with contemporary neural representations. For context, the authors cited include Krishna Subramani, Paris Smaragdis, Takuya Higuchi, and Mehrez Souden. Apple Machine Learning Research article

What’s new

The central contribution is a formal reformulation of NMF in terms of learnable functions rather than fixed vector factors. By treating the factors as functions that can be parameterized (for example, by neural networks or other adaptable function classes), the method no longer requires the data to be arranged on a regular grid or stored as a conventional matrix. This functional perspective enables NMF to operate on irregularly sampled TF representations and other signal forms that previously fell outside the standard NMF framework. In practical terms, this means that IR (irregular) representations like Constant-Q, wavelet-based decompositions, or sinusoidal models can be incorporated into a factorization pipeline in a way that preserves the non-negativity constraint and the interpretability benefits of NMF. The approach is presented as a principled extension rather than a ad-hoc workaround, with the theoretical and experimental framing aimed at demonstrating feasibility across diverse signal classes that do not fit neatly into a matrix form.

Why it matters (impact for developers/enterprises)

Expanding NMF to irregular TF representations opens new avenues for audio analysis in production environments. For developers, this can translate into more flexible feature extraction pipelines that are aligned with domain-specific representations—such as musical pitch content captured by Constant-Q or time-frequency analyses tailored to perceptual scales—without sacrificing the interpretability or non-negativity guarantees that NMF provides. Enterprises working with large-scale audio data could benefit from more versatile decomposition capabilities in areas like source separation, audio restoration, music information retrieval, and acoustic scene analysis. By enabling NMF to operate on a broader set of representations, the approach reduces the need to force data into a matrix form that may not align with the signal’s intrinsic structure, potentially improving both performance and computational efficiency when dealing with non-standard TF representations.

Technical details or Implementation (high level)

• The key shift is to replace fixed-factor vectors with learnable functions that parametrize the NMF factors. This allows the model to adapt to the sampling structure of the signal rather than being constrained by a pre-defined matrix layout.
• By working in a function space, irregular sampling grids become natural inputs to the factorization process, enabling direct interaction with representations that do not tile into a conventional matrix.
• The formulation preserves the non-negativity principle central to NMF, while introducing learnable components that can capture structure across irregular time-frequency domains.
• While the public summary does not disclose all architectural specifics, the approach is described as a generalization of NMF that remains faithful to the original goal of parts-based decomposition in an extended signal class setting.

Key takeaways

• NMF can be generalized beyond matrices to learnable function representations, broadening its applicability to irregular TF representations.
• The approach targets signal classes that are not naturally regularly sampled, addressing a long-standing limitation in standard NMF.
• The work ties intoNeurIPS discourse and WASPAA 2025 expectations, illustrating cross-domain relevance in audio and signal processing.
• The authors reference a set of researchers including Krishna Subramani, Paris Smaragdis, Takuya Higuchi, and Mehrez Souden, signaling a collaboration between established experts in the field.
• The publication underscores the importance of aligning matrix-factorization techniques with modern learnable representations to support flexible audio analysis pipelines.

References

• https://machinelearning.apple.com/research/rethinking-non-negative