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

arXiv: https://arxiv.org/abs/2506.05617 · HTML: https://mlsanity.com/paper/2506.05617/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 28 references that are in the map.
- First posted on arXiv 5 Jun 2025.

## Builds on (most cited first, 19 of 19)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Exploiting Linear Structure Within Convolutional Networks for Efficient Evaluation](https://mlsanity.com/paper/1404.0736/) (2014)
- [Parseval Networks: Improving Robustness to Adversarial Examples](https://mlsanity.com/paper/1704.08847/) (2017)
- [Speeding up Convolutional Neural Networks with Low Rank Expansions](https://mlsanity.com/paper/1405.3866/) (2014)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks](https://mlsanity.com/paper/1802.04034/) (2018)
- [Language model compression with weighted low-rank factorization](https://mlsanity.com/paper/2207.00112/) (2022)
- [Spectral Norm Regularization for Improving the Generalizability of Deep Learning](https://mlsanity.com/paper/1705.10941/) (2017)
- [Accelerating Very Deep Convolutional Networks for Classification and Detection](https://mlsanity.com/paper/1505.06798/) (2015)
- [The Singular Values of Convolutional Layers](https://mlsanity.com/paper/1805.10408/) (2018)
- [GroupReduce: Block-Wise Low-Rank Approximation for Neural Language Model Shrinking](https://mlsanity.com/paper/1806.06950/) (2018)
- [Online Embedding Compression for Text Classification using Low Rank Matrix Factorization](https://mlsanity.com/paper/1811.00641/) (2018)
- [MGIC: Multigrid-in-Channels Neural Network Architectures](https://mlsanity.com/paper/2011.09128/) (2020)
- [The SVD of Convolutional Weights: A CNN Interpretability Framework](https://mlsanity.com/paper/2208.06894/) (2022)
- [Invisible Backdoor Attack Through Singular Value Decomposition](https://mlsanity.com/paper/2403.13018/) (2024)
- [Poly-MgNet: Polynomial Building Blocks in Multigrid-Inspired ResNets](https://mlsanity.com/paper/2503.10594/) (2025)

## Abstract (arXiv)

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.