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This paper investigates the theoretical problem of maintaining linearseparability of the data-generating distribution under linear compression.While it has been long known that linear separability may be maintained bylinear transformations that approximately preserve the inner products betweenthe domain points, the limit to which the inner products are preserved in orderto maintain linear separability was unknown. In this paper, we show that linearseparability is maintained as long as the distortion of the inner products issmaller than the squared margin of the original data-generating distribution.The proof is mainly based on the geometry of hard support vector machines (SVM)extended from the finite set of training examples to the (possibly) infinitedomain of the data-generating distribution. As applications, we derive boundson the (i) compression length of random sub-Gaussian matrices; and (ii)generalization error for compressive learning with hard-SVM.

On Linear Separability under Linear Compression with Applications to Hard Support Vector Machine