On sharp stable recovery from clipped and folded measurements

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.