Clebsch-Gordan CNN for Spherical Data in Fourier Space
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Solution Overview
Problem
Existing convolutional neural networks (CNNs) face computational challenges when dealing with data defined on a sphere, particularly due to the need for repeated forward and backward Fourier transforms, which are costly and prone to numerical errors, limiting their efficiency and practicality in applications like image recognition and molecular chemistry.
Innovation Solution
The proposed Clebsch-Gordan CNN architecture operates entirely in Fourier space, using Clebsch-Gordan transforms to compute activations and avoid inverse and forward Fourier transforms across layers, maintaining computations in Fourier space to reduce computational costs and enhance efficiency.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If repeated forward and backward Fourier transforms are used in spherical CNNs, then equivariance to rotations is maintained, but computational cost increases and numerical errors occur
Solution Approach 1:
The patent extracts the nonlinear activation operation from the Fourier transform cycle by designing a fully Fourier-space architecture where activations are computed directly in Fourier space using Clebsch-Gordan coefficients, eliminating the need to transform back to real space for nonlinear operations
Solution Approach 2:
The patent segments the convolution operation into distinct Fourier space components using Clebsch-Gordan decomposition, allowing each layer to process features independently in Fourier space without requiring repeated transforms between domains
2Reliability
If repeated forward and backward Fourier transforms are used in spherical CNNs, then equivariance to rotations is maintained, but computational efficiency decreases
Solution Approach 1:
The patent maintains continuous operation in Fourier space across all network layers, allowing the useful computational action to proceed without interruption from transform cycles, thereby improving overall computational efficiency while preserving rotational equivariance
Solution Approach 2:
The patent performs an initial Fourier transform of the input data before processing, and then all subsequent operations including nonlinear activations are performed in Fourier space, eliminating the need for repeated transform pairs throughout the network
3Reliability
If repeated forward and backward Fourier transforms are used in spherical CNNs, then rotational equivariance is ensured, but numerical errors increase
Solution Approach 1:
The patent extracts the source of numerical errors by removing the repeated Fourier transform operations from the network architecture, computing activations directly in Fourier space using analytically derived Clebsch-Gordan coefficients to avoid transform-related precision loss
Data Source
AI summary
Methods and systems for computationally processing data with a multi-layer convolutional neural network (CNN) having an input and output layer, and one or more intermediate layers are described. Input data represented in a form of evaluations of continuous functions on a sphere may be received at a computing device and input to the input layer. The input layer may compute outputs as covariant Fourier space activations by transforming the continuous functions into spherical harmonic expansions. The output activations from the input layer may be processed sequentially through each of the intermediate layers. Each, intermediate layer may apply Ciebsch-Gordan transforms to compute respective covariant Fourier space activations as input to an immediately next layer, without computing any intermediate inverse Fourier transforms or forward Fourier transforms. Finally, the respective covariant Fourier space activations of the last intermediate layer may be processed in the output layer of the CNN to compute invariant activations.