LFA applied to CNNs: Efficient Singular Value Decomposition of Convolutional Mappings by Local Fourier Analysis

TL;DR

This paper introduces a new efficient method based on local Fourier analysis to compute all singular values of convolutional mappings, significantly reducing computational complexity compared to FFT-based approaches, enabling scalable spectral analysis of CNNs.

Contribution

The paper presents a novel O(N) complexity algorithm for singular value computation of convolutional mappings using local Fourier analysis, improving scalability over existing FFT-based methods.

Findings

The proposed method is scalable for high-dimensional convolutions.

It accurately computes the entire set of singular values and vectors.

Numerical experiments validate the efficiency and practicality of the approach.

Abstract

The singular values of convolutional mappings encode interesting spectral properties, which can be used, e.g., to improve generalization and robustness of convolutional neural networks as well as to facilitate model compression. However, the computation of singular values is typically very resource-intensive. The naive approach involves unrolling the convolutional mapping along the input and channel dimensions into a large and sparse two-dimensional matrix, making the exact calculation of all singular values infeasible due to hardware limitations. In particular, this is true for matrices that represent convolutional mappings with large inputs and a high number of channels. Existing efficient methods leverage the Fast Fourier transformation (FFT) to transform convolutional mappings into the frequency domain, enabling the computation of singular values for matrices representing convolutions with larger input and channel dimensions. For a constant number of channels in a given convolution, an FFT can compute N singular values in O(N log N) complexity. In this work, we propose an approach of complexity O(N) based on local Fourier analysis, which additionally exploits the shift invariance of convolutional operators. We provide a theoretical analysis of our algorithm's runtime and validate its efficiency through numerical experiments. Our results demonstrate that our proposed method is scalable and offers a practical solution to calculate the entire set of singular values - along with the corresponding singular vectors if needed - for high-dimensional convolutional mappings.