Replicating Kernels with a Short Stride Allows Sparse Reconstructions\n with Fewer Independent Kernels

Abstract

In sparse coding it is common to tile an image into nonoverlapping patches,\nand then use a dictionary to create a sparse representation of each tile\nindependently. In this situation, the overcompleteness of the dictionary is the\nnumber of dictionary elements divided by the patch size. In deconvolutional\nneural networks (DCNs), dictionaries learned on nonoverlapping tiles are\nreplaced by a family of convolution kernels. Hence adjacent points in the\nfeature maps (V1 layers) have receptive fields in the image that are\ntranslations of each other. The translational distance is determined by the\ndimensions of V1 in comparison to the dimensions of the image space. We refer\nto this translational distance as the stride.\n We implement a type of DCN using a modified Locally Competitive Algorithm\n(LCA) to investigate the relationship between the number of kernels, the\nstride, the receptive field size, and the quality of reconstruction. We find,\nfor example, that for 16x16-pixel receptive fields, using eight kernels and a\nstride of 2 leads to sparse reconstructions of comparable quality as using 512\nkernels and a stride of 16 (the nonoverlapping case). We also find that for a\ngiven stride and number of kernels, the patch size does not significantly\naffect reconstruction quality. Instead, the learned convolution kernels have a\nnatural support radius independent of the patch size.\n