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

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.

Poly-MgNet: Polynomial Building Blocks in Multigrid-Inspired ResNets

The structural analogies of ResNets and Multigrid (MG) methods such as common building blocks like convolutions and poolings where already pointed out by He et al.\ in 2016. Multigrid methods are used in the context of scientific computing for solving large sparse linear systems arising from partial differential equations. MG methods particularly rely on two main concepts: smoothing and residual restriction / coarsening. Exploiting these analogies, He and Xu developed the MgNet framework, which integrates MG schemes into the design of ResNets. In this work, we introduce a novel neural network building block inspired by polynomial smoothers from MG theory. Our polynomial block from an MG perspective naturally extends the MgNet framework to Poly-Mgnet and at the same time reduces the number of weights in MgNet. We present a comprehensive study of our polynomial block, analyzing the choice of initial coefficients, the polynomial degree, the placement of activation functions, as well as of batch normalizations. Our results demonstrate that constructing (quadratic) polynomial building blocks based on real and imaginary polynomial roots enhances Poly-MgNet's capacity in terms of accuracy. Furthermore, our approach achieves an improved trade-off of model accuracy and number of weights compared to ResNet as well as compared to specific configurations of MgNet.

MGiaD: Multigrid in all dimensions. Efficiency and robustness by coarsening in resolution and channel dimensions

Current state-of-the-art deep neural networks for image classification are made up of 10 - 100 million learnable weights and are therefore inherently prone to overfitting. The complexity of the weight count can be seen as a function of the number of channels, the spatial extent of the input and the number of layers of the network. Due to the use of convolutional layers the scaling of weight complexity is usually linear with regards to the resolution dimensions, but remains quadratic with respect to the number of channels. Active research in recent years in terms of using multigrid inspired ideas in deep neural networks have shown that on one hand a significant number of weights can be saved by appropriate weight sharing and on the other that a hierarchical structure in the channel dimension can improve the weight complexity to linear. In this work, we combine these multigrid ideas to introduce a joint framework of multigrid inspired architectures, that exploit multigrid structures in all relevant dimensions to achieve linear weight complexity scaling and drastically reduced weight counts. Our experiments show that this structured reduction in weight count is able to reduce overfitting and thus shows improved performance over state-of-the-art ResNet architectures on typical image classification benchmarks at lower network complexity.