Error bounds for consistent reconstruction: random polytopes and coverage processes

Consistent reconstruction is a method for producing an estimate x̃ ∈ R of a signal x ∈ R if one is given a collection ofN noisy linear measurements qn = 〈x, φn〉+ǫn, 1 ≤ n ≤ N , that have been corrupted by i.i.d. uniform noise {ǫn}n=1. We prove mean squared error bounds for consistent reconstruction when the measurement vectors {φn}n=1 ⊂ R are drawn independently at random from a suitable distribution on the unit-sphere S. Our main results prove that the mean squared error (MSE) for consistent reconstruction is of the optimal order E‖x − x̃‖2 ≤ Kδ/N under general conditions on the measurement vectors. We also prove refined MSE bounds when the measurement vectors are i.i.d. uniformly distributed on the unit-sphere S and, in particular, show that in this case the constant K is dominated by d, the cube of the ambient dimension. The proofs involve an analysis of random polytopes using coverage processes on the sphere.