Herglotz-NET: Implicit Neural Representation of Spherical~Data with Harmonic Positional Encoding

2025-02-19

AI Summary

Spherical data modeling suffers from inaccuracies and numerical instability when applying Euclidean methods due to intrinsic curvature. To address this, we propose the first harmonic positional encoding for implicit neural representations (INRs) on the sphere, grounded in complex Herglotz wave mappings—yielding a well-posed, interpretable, and spectrally robust formulation. We further develop a unified expressivity analysis framework that theoretically quantifies, for the first time, the relationship between network depth and the order of spherical harmonic expansion. By synergistically integrating spherical harmonic analysis with spectral-theoretic modeling, our method achieves significant improvements over existing INR approaches in reconstruction accuracy, numerical stability, and spectral controllability. The resulting paradigm is scalable, highly flexible, and enables high-fidelity modeling of spherical data.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Representation Learning for Vision

Application Category

• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

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.

Problem

• Addresses spherical data representation challenges
• Adapts implicit neural representations for spheres
• Introduces harmonic encoding for spectral properties

Innovation

• Harmonic positional encoding
• Complex Herglotz mappings
• Unified expressivity analysis