用谐波编码解决球面数据的神经表示难题
Herglotz-NET: Implicit Neural Representation of Spherical Data with Harmonic Positional Encoding
- 基于复数Herglotz映射设计球面谐波位置编码
- 理论证明网络深度与频谱表达能力正相关
- 适合需要高精度建模球面数据的研究者
球面域的数据表示与处理面临独特挑战，主要源于其曲率特性，使得经典欧氏方法难以直接应用。隐式神经表示（INRs）作为高保真数据表示的有前途方案，但要在球面上有效应用，必须适应球体固有的几何结构以保证准确性和稳定性。本文提出Herglotz-NET（HNET），一种基于复数Herglotz映射的谐波位置编码的新型INR架构。该编码在球面上产生具有良好适定性、可解释且鲁棒的频谱性质。此外，我们给出一个统一的表达能力分析，表明任何满足弱条件的球面型INR均具有可预测的频谱展开，其表达能力随网络深度增长。结果确立了HNET作为可扩展、灵活的球面数据精确建模框架。
原文摘要 · Abstract (English)
Representing and processing data in spherical domains presents unique challenges, primarily due to the curvature of the domain, which complicates the application of classical Euclidean techniques. Implicit neural representations (INRs) have emerged as a promising alternative for high-fidelity data representation; however, to effectively handle spherical domains, these methods must be adapted to the inherent geometry of the sphere to maintain both accuracy and stability. In this context, we propose Herglotz-NET (HNET), a novel INR architecture that employs a harmonic positional encoding based on complex Herglotz mappings. This encoding yields a well-posed representation on the sphere with interpretable and robust spectral properties. Moreover, we present a unified expressivity analysis showing that any spherical-based INR satisfying a mild condition exhibits a predictable spectral expansion that scales with network depth. Our results establish HNET as a scalable and flexible framework for accurate modeling of spherical data.
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