MetricGrids: Arbitrary Nonlinear Approximation with Elementary Metric Grids based Implicit Neural Representation

2025-03-13

AI Summary

Existing grid-based implicit representations are constrained by linear latent spaces, limiting their capacity to efficiently model complex nonlinear signals. This paper proposes MetricGrids: first, constructing base grids over diverse metric spaces and combining higher-order terms to achieve Taylor-like nonlinear modeling; second, designing a sparsity-adaptive hash encoding scheme and a higher-order extrapolation decoder to enhance representational power while maintaining parameter efficiency. MetricGrids is the first approach to incorporate metric space structure into grid representations, thereby overcoming the linearity limitations inherent in conventional Euclidean grids. Experiments demonstrate that MetricGrids significantly improves fitting accuracy and generalization robustness across 2D/3D signal reconstruction and neural rendering tasks. By grounding implicit neural representations in geometrically meaningful metric structures, MetricGrids establishes a new paradigm with stronger theoretical foundations and enhanced geometric interpretability.

Technology Category

• Machine Learning: Representation Learning
• Computer Vision: Representation Learning for Vision
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

Abstract

This paper presents MetricGrids, a novel grid-based neural representation that combines elementary metric grids in various metric spaces to approximate complex nonlinear signals. While grid-based representations are widely adopted for their efficiency and scalability, the existing feature grids with linear indexing for continuous-space points can only provide degenerate linear latent space representations, and such representations cannot be adequately compensated to represent complex nonlinear signals by the following compact decoder. To address this problem while keeping the simplicity of a regular grid structure, our approach builds upon the standard grid-based paradigm by constructing multiple elementary metric grids as high-order terms to approximate complex nonlinearities, following the Taylor expansion principle. Furthermore, we enhance model compactness with hash encoding based on different sparsities of the grids to prevent detrimental hash collisions, and a high-order extrapolation decoder to reduce explicit grid storage requirements. experimental results on both 2D and 3D reconstructions demonstrate the superior fitting and rendering accuracy of the proposed method across diverse signal types, validating its robustness and generalizability. Code is available at https://github.com/wangshu31/MetricGrids}{https://github.com/wangshu31/MetricGrids.

Problem

• Approximates complex nonlinear signals using elementary metric grids.
• Enhances grid-based neural representations with high-order terms.
• Reduces storage needs with hash encoding and extrapolation decoder.

Innovation

• Combines elementary metric grids for nonlinear signals
• Uses hash encoding to enhance model compactness
• Employs high-order extrapolation decoder for storage efficiency