Universal 1-Bit Compressive Sensing for Bounded Dynamic Range Signals

TL;DR

This paper establishes bounds on the number of measurements needed for universal 1-bit compressive sensing to recover support of sparse signals with bounded dynamic range, highlighting the impact of dynamic range and sparsity.

Contribution

It provides the first analysis of support recovery bounds for 1-bit CS on signals with bounded dynamic range, showing the dependence on dynamic range and sparsity.

Findings

Lower bound of rac{ ilde{ ext{O}}(Rk^{3/2})}{ ext{measurements}} for support recovery

Upper bound of rac{O(Rk^{3/2} ext{log} n)}{ ext{measurements}} matching the lower bound up to logarithmic factors

Support recovery is more measurement-efficient for signals with bounded dynamic range compared to arbitrary sparse signals.

Abstract

A {\em universal 1-bit compressive sensing (CS)} scheme consists of a measurement matrix $A$ such that all signals $x$ belonging to a particular class can be approximately recovered from $\textrm{sign}(Ax)$. 1-bit CS models extreme quantization effects where only one bit of information is revealed per measurement. We focus on universal support recovery for 1-bit CS in the case of {\em sparse} signals with bounded {\em dynamic range}. Specifically, a vector $x \in \mathbb{R}^n$ is said to have sparsity $k$ if it has at most $k$ nonzero entries, and dynamic range $R$ if the ratio between its largest and smallest nonzero entries is at most $R$ in magnitude. Our main result shows that if the entries of the measurement matrix $A$ are i.i.d.~Gaussians, then under mild assumptions on the scaling of $k$ and $R$, the number of measurements needs to be $\tilde{\Omega}(Rk^{3/2})$ to recover the support of $k$-sparse signals with dynamic range $R$ using $1$-bit CS. In addition, we show that a near-matching $O(R k^{3/2} \log n)$ upper bound follows as a simple corollary of known results. The $k^{3/2}$ scaling contrasts with the known lower bound of $\tilde{\Omega}(k^2 \log n)$ for the number of measurements to recover the support of arbitrary $k$-sparse signals.