Sparse recovery on sphere via probabilistic compressed sensing

It is difficult to determine whether or not the restricted isometry property (RIP) holds when measurements are taken on a given order. Hence, a probabilistic and RIPless compressed sensing that requires weaker and simpler conditions was recently developed. However, in unbounded orthonormal systems such as spherical harmonics, this theory on its own does not yield an optimum bound on the minimum number of required measurements. This is primarily due to the coherence of spherical harmonics growing with the band-limit and varying with the position of sample points. In this paper, we incorporate a preconditioning technique into the probabilistic approach to derive a slightly improved bound on the order of measurements for accurate recovery of spherical harmonic expansions.