Implicit Convolutional Kernels for Steerable CNNs (2212.06096v3)

Abstract: Steerable convolutional neural networks (CNNs) provide a general framework for building neural networks equivariant to translations and transformations of an origin-preserving group $G$, such as reflections and rotations. They rely on standard convolutions with $G$-steerable kernels obtained by analytically solving the group-specific equivariance constraint imposed onto the kernel space. As the solution is tailored to a particular group $G$, implementing a kernel basis does not generalize to other symmetry transformations, complicating the development of general group equivariant models. We propose using implicit neural representation via multi-layer perceptrons (MLPs) to parameterize $G$-steerable kernels. The resulting framework offers a simple and flexible way to implement Steerable CNNs and generalizes to any group $G$ for which a $G$-equivariant MLP can be built. We prove the effectiveness of our method on multiple tasks, including N-body simulations, point cloud classification and molecular property prediction.

• The paper presents an implicit kernel parameterization that automatically enforces G-equivariance without deriving group-specific kernel bases.
• It leverages MLPs designed to be G-equivariant, simplifying the architecture while extending applicability to any origin-preserving transformation group.
• Empirical tests in N-body simulations, point cloud classification, and molecular predictions show enhanced flexibility and performance over traditional methods.

An Analysis of "Implicit Convolutional Kernels for Steerable CNNs"

The paper presents a novel approach to designing Steerable Convolutional Neural Networks (CNNs) by using implicit convolutional kernels. The authors propose parameterizing $G$-steerable kernels through implicit neural representations via multi-layer perceptrons (MLPs), offering a unified framework that extends the applicability of steerable CNNs to any origin-preserving transformation group $G$. This method provides a flexible alternative to classical solutions, which usually require analytically deriving a kernel basis for each specific group.

Technical Summary

Steerable CNNs are inherently designed to be equivariant to transformations in a compact group $G$, ensuring the network’s responses change predictably when the input is transformed. Traditional approaches involve deriving $G$-steerable kernels by solving the equivariance constraint explicitly for each group $G$. The primary limitation of this approach is the lack of generalizability across different symmetry transformations, as the solution is deeply tailored to a specific group.

The authors tackle this challenge by employing implicit neural representations to parameterize steerable kernels, which leverage MLPs that themselves are $G$-equivariant. By ensuring that the MLPs maintain equivariance through appropriately designed representations of the group actions, they facilitate automatic compliance with the equivariance constraint, thereby removing the need for group-specific kernel solutions. This method not only broadens the scope of symmetry transformations that steerable CNNs can achieve but also simplifies the implementation process.

Empirical Results and Insights

The framework’s effectiveness is demonstrated across multiple domains including N-body simulations, point cloud classification, and molecular property prediction tasks. The empirical results highlight the flexibility and superior performance of these implicit kernels compared to traditional approaches, especially when dealing with arbitrary sub-groups of the orthogonal group $O(n)$.

• N-body Simulations and Point Clouds: The experiments underline implicit kernels' adaptability in handling datasets with complex symmetry requirements, illustrative of the N-body simulation task. Here, the implicit kernels effectively capture the underlying physical dynamics, reflecting enhancements over baseline models.
• Molecular Property Predictions: In molecular data prediction tasks, the inclusion of structural and interaction information directly into kernel computations showcased marked improvements over other models, attributing to the flexible input handling of MLP-based kernels.

Theoretical and Practical Implications

The introduction of implicit convolutional kernels marks a significant theoretical advancement, suggesting a shift in how equivariance is achieved in neural architectures. By facilitating a more generalized formulation of steerable convolutions, the paper paves the way for broad applicative potential, extending into domains where pre-defined symmetry transformations might inhibit the use of conventional steerable CNNs.

Practically, this flexibility allows researchers and developers to incorporate more complex symmetries and domain-specific features directly into the learning processes. Increased expressivity and model performance, as observed, are critical for advancing machine learning solutions in physical simulations, computer vision, and beyond.

Future Directions

The paper opens up several avenues for future research. Opportunities include optimizing the depth and architecture of the MLPs used for kernel parameterization, improving computational efficiency, and exploring additional symmetry groups beyond the orthogonal transformations currently considered. Furthermore, extending this approach to newer data modalities and tasks (e.g., 4D data processing) could harness the full potential of steerable CNNs optimized through implicit kernels.

In conclusion, the authors present evidence that implicit neural representations are not just a viable alternative but an advantageous enhancement for the design and deployment of equivariant neural networks, significantly broadening the horizons for research and application in equivariant deep learning.