Rohit Pochampalli, Chair for Scientific Computing (SciComp), TU Kaiserslautern
Title: Scale Separation in Convolutional Neural Networks: A Multigrid Approach
The remarkable performance of convolutional neural networks on computer vision tasks is closely linked to the properties of the convolutional filter, especially by means of the imposition of geometric priors, translational equivariance and scale separation on the neural network. By virtue of these properties, convolutional filters, and thus in turn convolutional neural networks, are able to exploit the geometric structure and symmetries of the underlying domain (which is the Hilbert space of image representations). Considering the example of image classification, scale separation emerges from the preservation of important characteristics of the image concurrent with subsampling and coarse-graining that occur as the image is propagated through the layers of the network. As a consequence, convolutional neural networks benefit from a separation of important characteristics of the image across several scales.
A similar principle arises in the multigrid scheme, used to speed up the numerical solution of partial differential equations. Here, an iterative method is separated into several levels that smoothen errors on different frequency bands. In this work we look at a novel characterization of scale separation within layers of a convolutional neural network. This is achieved by introducing neural network architectures that utilize ideas from the multigrid method. Explicitly, a two-grid multigrid cycle is incorporated into a convolutional layer analogous to the restriction and prolongation operations of the multigrid approach. The use of dilation in the convolution operations is shown to enable the representation of image features in multiple scales of abstraction, conducive to the extraction of relevant morphological structures across these scales.
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