Error Bounds for Consistent Reconstruction: Random Polytopes and Coverage Processes

Abstract

Consistent reconstruction is a method for producing an estimate $\widetilde{x} \in {\mathbb {R}}^d$ of a signal $x\in {\mathbb {R}}^d$ if one is given a collection of $N$ noisy linear measurements $q_n = \langle x, \varphi _n \rangle + \epsilon _n$, $1 \le n \le N$, that have been corrupted by i.i.d. uniform noise $\{\epsilon _n\}_{n=1}^N$. We prove mean-squared error bounds for consistent reconstruction when the measurement vectors $\{\varphi _n\}_{n=1}^N\subset {\mathbb {R}}^d$ are drawn independently at random from a suitable distribution on the unit-sphere ${\mathbb {S}}^{d-1}$. Our main results prove that the mean-squared error (MSE) for consistent reconstruction is of the optimal order ${\mathbb {E}}\Vert x - \widetilde{x}\Vert ^2 \le K\delta ^2/N^2$ 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 ${\mathbb {S}}^{d-1}$ and, in particular, show that in this case, the constant $K$ is dominated by $d^3$, the cube of the ambient dimension. The proofs involve an analysis of random polytopes using coverage processes on the sphere.