In this paper, we consider the downlink of a cognitive radio network where a cognitive base station serves multiple cognitive users on the same frequency band as a group of primary transceivers. The cognitive base station uses an orthogonal scheduling scheme (TDMA/FDMA) to serve its users. For this purpose, the base station is interested in acquiring an estimate of the interference (from the primary network) power at each of its cognitive receivers as a measure of channel quality. This can be surely achieved if we allow for the feedback (from the cognitive receivers to the cognitive base station) bandwidth to scale linearly in the number of cognitive receivers, but in densely populated networks, the cost of such an acquisition might be too high. This leads us to the question of whether we can do better in terms of bandwidth efficiency. We observe that in many scenarios - that are common in practice - where the primary network exhibits sparse changes in transmit powers from one scheduling instant to the next, it is possible to acquire this interference state with only a logarithmic scaling in feedback bandwidth. More specifically, in cognitive networks where the channels are solely determined by the positions of nodes, we can use compressed sensing to recover the interference state. In addition to being a first application of compressed sensing in the domain of limited feedback, to the best of our knowledge, this paper makes a key mathematical contribution concerning the favourable sensing properties of path-loss matrices that are composed of nonzero mean, dependent random entries. Finally, we numerically study the robustness properties of the least absolute shrinkage and selection operator (LASSO), a popular recovery algorithm, under two error models through simulations. The first model considers a varying amount of error added to all entries of the sensing matrix. The second one, a more adversarial model, considers a large amount of error added to only a fraction of the entries of the sensing matrix that are chosen uniformly at random. Simulation results establish that the LASSO recovery algorithm is robust to imperfect channel knowledge.