Grounding and Enhancing Grid-based Models for Neural Fields

Abstract: Many contemporary studies utilize grid-based models for neural field representation, but a systematic analysis of grid-based models is still missing, hindering the improvement of those models. Therefore, this paper introduces a theoretical framework for grid-based models. This framework points out that these models' approximation and generalization behaviors are determined by grid tangent kernels (GTK), which are intrinsic properties of grid-based models. The proposed framework facilitates a consistent and systematic analysis of diverse grid-based models. Furthermore, the introduced framework motivates the development of a novel grid-based model named the Multiplicative Fourier Adaptive Grid (MulFAGrid). The numerical analysis demonstrates that MulFAGrid exhibits a lower generalization bound than its predecessors, indicating its robust generalization performance. Empirical studies reveal that MulFAGrid achieves state-of-the-art performance in various tasks, including 2D image fitting, 3D signed distance field (SDF) reconstruction, and novel view synthesis, demonstrating superior representation ability. The project website is available at https://sites.google.com/view/cvpr24-2034-submission/home.

• The paper proposes MulFAGrid, which leverages Grid Tangent Kernels to link grid architecture with training dynamics for improved generalization.
• MulFAGrid combines multiplicative filtering with Fourier features to efficiently capture high-frequency components in neural fields.
• Empirical results demonstrate that MulFAGrid outperforms earlier grid-based models in tasks like 2D image fitting, 3D SDF reconstruction, and novel view synthesis.

Enhancements in Grid-Based Models for Neural Fields through MulFAGrid: A Theoretical and Empirical Perspective

Introduction

Grid-based models have demonstrated considerable success in various tasks involving neural fields, such as 2D image fitting, 3D signed distance field (SDF) reconstruction, and novel view synthesis. The efficiency and fidelity of these models, which represent continuous entities over grids and leverage grid feature tensors, have been empirically validated against MLP-based counterparts. However, the theoretical foundations explaining the behaviors and efficacies of grid-based models remain underexplored. Addressing this gap, recent efforts introduce a theoretical framework centered around Grid Tangent Kernels (GTKs), proposing a novel model—Multiplicative Fourier Adaptive Grid (MulFAGrid)—and showcasing its superior performance through extensive empirical studies.

Theoretical Framework

The Basis of Grid Tangent Kernels (GTKs)

GTKs serve as the cornerstone of the proposed theoretical framework for grid-based models. Analogous to neural tangent kernels (NTKs) in MLPs, GTKs quantify how parameter adjustments influence model predictions, offering insights into the optimization trajectories and generalization capabilities of grid-based models. The intrinsic property of GTKs, maintaining constancy throughout training, allows the interpretation of grid-based models' behavior similarly to linear kernelized models. A pivotal aspect of this framework is its ability to directly relate the architecture of grid-based models to their training dynamics and generalization through GTK analysis.

Generalization Performance

By establishing a connection between GTK spectra and generalization bounds, the framework elucidates factors critical to the models' ability to generalize from trained data to unseen scenarios. Empirical studies concur with the theoretical predictions that models designed based on optimizing the GTK spectrum could achieve robust generalization capabilities.

The Multiplicative Fourier Adaptive Grid (MulFAGrid)

Architectural Innovations

Inspired by the theoretical insights from GTKs, MulFAGrid utilizes multiplicative filters and Fourier features to develop a dynamic grid-based model adaptable to both regular and irregular grids. By incorporating an adaptive learning scheme for simultaneous optimization of kernel features and grid features, MulFAGrid outperforms its predecessors. Specifically, the model demonstrates an expansive GTK spectrum, particularly in the high-frequency domain, facilitating efficient learning of high-frequency components.

Empirical Validation

Extensive experiments conducted on canonical tasks involving neural fields reveal MulFAGrid's exceptional capability in fitting 2D images and reconstructing 3D SDFs, alongside novel view synthesis with state-of-the-art performance. Notably, MulFAGrid performs competitively against other advanced grid-based models while showcasing superior fidelity and efficiency.

Implications and Prospects

The advent of the MulFAGrid model, underpinned by the GTK-based theoretical framework, heralds significant advancements in understanding and innovating grid-based models for neural fields. The remarkable performance of MulFAGrid across a spectrum of applications not only validates the theoretical findings but also sets a new benchmark for future explorations. Reflecting on the profound implications of this research, a few pathways for further studies emerge:

• Theoretical Expansions: The static nature of GTKs during training presents an avenue to explore the theoretical aspects of grid-based models, possibly extending the framework to encompass dynamic model behaviors over prolonged training periods or under different scenarios.
• Algorithmic Developments: The adaptive learning strategy in MulFAGrid opens up opportunities for developing more sophisticated algorithms that could further enhance the learning efficiency and generalization performance of grid-based models.
• Application Horizons: Given MulFAGrid’s adaptability to both regular and irregular grids, investigating its application across a wider range of domains beyond the current scope could reveal its full potential and versatility.

The groundbreaking theoretical and empirical contributions of this research not only fortify the understanding of grid-based models but also pave the way for innovative developments in the field of neural fields.