This paper studies the classical problem of communication across channels with state estimated at the receiver, in the context of wireless channels with reconfigurable intelligent surfaces (RIS). The RIS channels are characterized by numerous channel parameters but often have a sparse underlying structure. Under these conditions, the communication, channel training, and the characteristics of sparse recovery algorithms, interact in intricate ways. We calculate a training-based achievable rate for the RIS-induced sparse channel. We use an efficient sparse model for the RIS-aided channel that eliminates the need for recovering angles of arrival and departure at the RIS. We incorporate in the analysis the misalignment between the discrete parameter model of compressive sensing and the actual continuous-valued channel parameters, referred to as basis mismatch. Finally, we offer insights into designing RIS size and compressive sensing-based channel estimation parameters for RIS-aided communication systems.