Sparse Recovery with Brownian Sensing

We consider the problem of recovering the parameter alpha in R^K of a sparse function f, i.e. the number of non-zero entries of alpha is small compared to the number K of features, given noisy evaluations of f at a set of well-chosen sampling points. We introduce an additional randomisation process, called Brownian sensing, based on the computation of stochastic integrals, which produces a Gaussian sensing matrix, for which good recovery properties are proven independently on the number of sampling points N, even when the features are arbitrarily non-orthogonal. Under the assumption that f is Hölder continuous with exponent at least 1/2, we provide an estimate a of the parameter such that ||\alpha - a||_2 = O(||eta||_2\sqrt{N}), where eta is the observation noise. The method uses a set of sampling points uniformly distributed along a one-dimensional curve selected according to the features. We report numerical experiments illustrating our method.