Error bounds for consistent reconstruction: random polytopes and coverage processes

TL;DR

This paper establishes optimal error bounds for consistent signal reconstruction from noisy measurements using random polytopes and coverage processes, demonstrating that the mean squared error decreases quadratically with the number of measurements.

Contribution

It provides the first rigorous mean squared error bounds for consistent reconstruction with random measurement vectors on the sphere, including refined bounds for uniform distributions.

Findings

Mean squared error is of order ^3/N^2 for uniform sphere measurements.

Error bounds are optimal and depend on the ambient dimension and number of measurements.

Analysis involves geometric properties of random polytopes and sphere 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 $q_n = \langle x, \varphi_n \rangle + \epsilon_n$, $1 \leq n \leq N$, 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 $\mathbb{E}\|x - \widetilde{x}\|^2 \leq K\delta^2/N^2$ 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.