Cognitive radio aims at identifying unused radio-frequency (RF) bands with the goal of re-using them opportunistically for other services. While compressive sensing (CS) has been used to identify strong signals (or interferers) in the RF spectrum from sub-Nyquist measurements, identifying unused frequencies from CS measurements appears to be uncharted territory. In this paper, we propose a novel method for identifying unused RF bands using an algorithm we call least matching pursuit (LMP). We present a sufficient condition for which LMP is guaranteed to identify unused frequency bands and develop an improved algorithm that is inspired by our theoretical result. We perform simulations for a CS-based RF whitespace detection task in order to demonstrate that LMP is able to outperform black-box approaches that build on deep neural networks.