LFA applied to CNNs: Efficient Singular Value Decomposition of Convolutional Mappings by Local Fourier Analysis
This work provides a practical solution for improving generalization, robustness, and model compression in CNNs by enabling efficient spectral analysis, though it is incremental as it builds on existing frequency-domain methods.
The paper tackles the problem of efficiently computing singular values for convolutional mappings in CNNs, which is typically resource-intensive, by proposing a method based on local Fourier analysis that achieves O(N) complexity and demonstrates scalability in numerical experiments.
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.