On sharp stable recovery from clipped and folded measurements

TL;DR

This paper establishes sharp bounds on the stability of recovering vectors from clipped or folded measurements, relevant for measurement devices with limited detection ranges, extending one-bit compressed sensing results.

Contribution

It provides the first sharp lower bounds on recovery constants for declipping and unfolding, under optimal sampling and data distribution conditions, with sparse and sphere-restricted extensions.

Findings

Sharp lower bounds on recovery constants for declipping and unfolding.

Effective bounds under optimal sample size and data distribution.

Extension of one-bit compressed sensing results to declipping.

Abstract

We investigate the stability of vector recovery from random linear measurements which have been either clipped or folded. This is motivated by applications where measurement devices detect inputs outside of their effective range. As examples of our main results, we prove sharp lower bounds on the recovery constant for both the declipping and unfolding problems whenever samples are taken according to a uniform distribution on the sphere. Moreover, we show such estimates under (almost) the best possible conditions on both the number of samples and the distribution of the data. We then prove that all of the above results have suitable (effectively) sparse counterparts. In the special case that one restricts the stability analysis to vectors which belong to the unit sphere of $\mathbb{R}^n$, we show that the problem of declipping directly extends the one-bit compressed sensing results of Oymak-Recht and Plan-Vershynin.