Sparse Recovery with Brownian Sensing

We consider the problem of recovering the parameter α ∈ ℝK of a sparse function f (i.e. the number of non-zero entries of α 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 randomization 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 Holder continuous with exponent at least 1/2, we provide an estimate ᾶ of the parameter such that ∥α - ᾶ∥2 = O(∥η∥2/√N), where η 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.