发表机构
Tel Aviv University（特拉维夫大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究提出奇点空间这一用于信号表示的生成扩散框架，基于复平面奇点，利用基于变压器的扩散模型，具有可解释性、结构稳定性等特性，在一维伯格斯激波测试中表现良好，为瞬变主导信号提供实用基础。
AI 中文摘要
生成模型常将信号表示为密集的幅度网格，模糊了对物理信号正确性至关重要的尖锐瞬变。我们引入奇点空间，这是一个通过复平面奇点表示信号的生成框架，源于亚纯函数的经典极点 - 留数表示。我们学习物理约束的、每个信号的奇点配置的潜在空间，以从退化或部分观测中解决逆问题。该框架有三个关键特性：可解释性，每个生成的奇点配置对应一组物理参数；结构稳定性，减轻不连续处的吉布斯伪影；无需重新训练或插值即可在任意网格上进行无分辨率输出重建。我们的框架采用基于变压器的扩散模型，直接预测复平面奇点坐标处的样本，采样时受几何约束。作为尖锐特征恢复的受控测试案例，我们在一维伯格斯激波上评估框架，每个激波由32个预测奇点表示（与1024点网格信号相比减少8倍）。我们的框架在未见测试时观测噪声下保留信号结构（总变差比≈1），在零样本子分辨率泛化中比基于网格的基线实现低4.2倍的重建误差，在分布内将物理参数恢复到绝对误差为10⁻⁴。这些结果表明基于奇点的表示可为语音和生物医学信号等其他瞬变主导信号提供实用基础，并可能扩展到高维领域。
英文摘要
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.
