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Motivated by the problem of learning with small sample sizes, this papershows how to incorporate into support-vector machines (SVMs) those propertiesthat have made convolutional neural networks (CNNs) successful. Particularlyimportant is the ability to incorporate domain knowledge of invariances, e.g.,translational invariance of images. Kernels based on the \textit{maximum}similarity over a group of transformations are not generally positive definite.Perhaps it is for this reason that they have not been studied theoretically. Weaddress this lacuna and show that positive definiteness indeed holds\textit{with high probability} for kernels based on the maximum similarity inthe small training sample set regime of interest, and that they do yield thebest results in that regime. We also show how additional properties such astheir ability to incorporate local features at multiple spatial scales, e.g.,as done in CNNs through max pooling, and to provide the benefits of compositionthrough the architecture of multiple layers, can also be embedded into SVMs. Weverify through experiments on widely available image sets that the resultingSVMs do provide superior accuracy in comparison to well-established deep neuralnetwork benchmarks for small sample sizes.

Learning from Few Samples: Transformation-Invariant SVMs with Composition and Locality at Multiple Scales