{"ID":6536300,"CreatedAt":"2026-07-14T01:21:01.169441415Z","UpdatedAt":"2026-07-15T03:28:55.185153975Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.10930","arxiv_id":"2607.10930","title":"The Singularity Space: A Generative Diffusion Framework for Signal Representation","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.","short_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 representat...","url_abs":"https://arxiv.org/abs/2607.10930","url_pdf":"https://arxiv.org/pdf/2607.10930v1","authors":"[\"Eli Bar-Yosef\",\"Amir Averbuch\",\"Eli Turkel\"]","published":"2026-07-12T21:34:13Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"cs.AI\",\"eess.SP\",\"math.NA\"]","methods":"[\"Diffusion Model\",\"Transformer\"]","has_code":false}