Sigma-Delta quantization of sub-Gaussian frame expansions and its application to compressed sensing

TL;DR

This paper demonstrates that signals can be accurately reconstructed from Sigma-Delta quantized sub-Gaussian frame expansions with error decreasing rapidly as oversampling increases, extending previous results to more general settings.

Contribution

It establishes root-exponential and polynomial error decay bounds for Sigma-Delta quantized sub-Gaussian frames, broadening the scope of compressed sensing and frame expansion analysis.

Findings

Error decays root-exponentially with oversampling rate

Polynomial error decay for fine quantization in compressed sensing

High probability bounds on singular values of sub-Gaussian frame matrices

Abstract

Suppose that the collection $\{e_i\}_{i=1}^m$ forms a frame for $\R^k$, where each entry of the vector $e_i$ is a sub-Gaussian random variable. We consider expansions in such a frame, which are then quantized using a Sigma-Delta scheme. We show that an arbitrary signal in $\R^k$ can be recovered from its quantized frame coefficients up to an error which decays root-exponentially in the oversampling rate $m/k$. Here the quantization scheme is assumed to be chosen appropriately depending on the oversampling rate and the quantization alphabet can be coarse. The result holds with high probability on the draw of the frame uniformly for all signals. The crux of the argument is a bound on the extreme singular values of the product of a deterministic matrix and a sub-Gaussian frame. For fine quantization alphabets, we leverage this bound to show polynomial error decay in the context of compressed sensing. Our results extend previous results for structured deterministic frame expansions and Gaussian compressed sensing measurements.