The Singularity Space: A Generative Diffusion Framework for Signal Representation

2026-07-12

AI Summary

Traditional generative models rely on dense grid representations, which struggle to accurately reconstruct sharp transient features in physical signals. This work proposes a generative diffusion framework based on singularities in the complex plane, achieving high-fidelity signal reconstruction from degraded or partial observations by learning singularity configurations that satisfy physical constraints. The approach introduces a structurally stable, resolution-independent, and interpretable singularity representation that effectively mitigates Gibbs artifacts and enables reconstruction on arbitrary grids. A Transformer-based diffusion model directly predicts complex-plane singularity coordinates consistent with geometric constraints. In Burgers' shock-wave tasks, the method surpasses the reconstruction accuracy of a 1024-point grid using only 32 singularities, reduces zero-shot sub-resolution generalization error by 4.2×, and achieves a physical parameter recovery error of 10⁻⁴.

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.

Problem

• signal representation
• sharp transients
• singularity
• generative models
• inverse problem

Innovation

• Singularity Space
• diffusion generative model
• pole-residue representation
• resolution-free reconstruction
• structured signal generation