The Singularity Space: A Generative Diffusion Framework for Signal Representation

• URL: http://arxiv.org/abs/2607.10930v1
• Date: Sun, 12 Jul 2026 21:34:13 GMT
• Title: The Singularity Space: A Generative Diffusion Framework for Signal Representation
• Abstract summary: We introduce Singularity Space, a generative framework that represents signals through complex-plane singularities.<n>We evaluate our framework on 1D Burgers, where each shock is represented by 32 predicted singularities.<n>Our framework preserves signal structure under unseen test-time observation noise, achieves a $4.2times$ lower reconstruction error in zero-shot sub-resolution generalization.
• Score: 0.8526086056172272
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Generative models often represent signals as dense grids of amplitudes, blurring sharp transients that are crucial for the correctness of physical signals. We introduce Singularity Space, a generative framework that represents signals through complex-plane singularities, rooted in the classical pole-residue representation of meromorphic functions. We learn a latent space of physically constrained, per-signal singularity configurations to solve an inverse problem from degraded or partial observations. The framework has three key properties: interpretability, in which each generated singularity configuration corresponds to a set of physical parameters; structural stability, which mitigates Gibbs artifacts at discontinuities; and resolution-free output reconstruction on arbitrary grids without retraining or interpolation. Our framework employs a transformer-based diffusion model that directly predicts samples at complex-plane singularity coordinates, subject to geometric constraints during sampling. As a controlled test case for sharp-feature recovery, we evaluate our framework on 1D Burgers shocks, where each shock is represented by 32 predicted singularities (an $8\times$ reduction versus a 1024-point grid signal). Our framework preserves signal structure ($\text{TV ratio} \approx 1$) under unseen test-time observation noise, achieves a $4.2\times$ lower reconstruction error in zero-shot sub-resolution generalization than a grid-based baseline, and recovers physical parameters to $10^{-4}$ absolute error in-distribution. These results suggest that singularity-based representations may provide a practical foundation for other transient-dominated signals such as speech and biomedical signals, with potential extension to higher-dimensional domains.

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