Using the kernel trick in compressive sensing: Accurate signal recovery from fewer measurements

Compressive sensing accurately reconstructs a signal that is sparse in some basis from measurements, generally consisting of the signal's inner products with Gaussian random vectors. The number of measurements needed is based on the sparsity of the signal, allowing for signal recovery from far fewer measurements than is required by the traditional Shannon sampling theorem. In this paper, we show how to apply the kernel trick, popular in machine learning, to adapt compressive sensing to a different type of sparsity. We consider a signal to be “nonlinearly K-sparse” if the signal can be recovered as a nonlinear function of K underlying parameters. Images that lie along a low-dimensional manifold are good examples of this type of nonlinear sparsity. It has been shown that natural images are as well [1]. We show how to accurately recover these nonlinearly K-sparse signals from approximately 2K measurements, which is often far lower than the number of measurements usually required under the assumption of sparsity in an orthonormal basis (e.g. wavelets). In experimental results, we find that we can recover images far better for small numbers of compressive sensing measurements, sometimes reducing the mean square error (MSE) of the recovered image by an order of magnitude or more, with little computation. A bound on the error of our recovered signal is also proved.