To make the system available at low-cost, millimeter-ave (mmWave) multiple-input multiple-output (MIMO) architectures employ analog arrays, which are driven by a limited number of radio frequency (RF) chains. One primary challenge of using large hybrid analog-digital arrays is that the digital baseband cannot directly access the signal to/from each antenna. To address this limitation, recent research has focused on retransmissions, iterative precoding, and subspace decomposition methods. Unlike these approaches that exploited the channel’s low-rank, in this work we exploit the sparsity of the received signal at both the transmit/receive antennas. While the signal itself is de facto dense, it is well-known that most signals are sparse under an appropriate choice of basis. By delving into the structured compressive sensing (CS) framework and adapting them to variants of the mmWave hybrid architectures, we provide methodologies to recover the analog signal at each antenna from the (low-dimensional) digital signal. Moreover, we characterizes the minimal numbers of measurement and RF chains to provide this recovery, with high probability. We discuss their applications to common variants of the hybrid architecture. By leveraging the inherent sparsity of the received signal, our analysis reveals that a hybrid MIMO system can be " turned into" a fully digital one: the number of needed RF chains increases logarithmically with the number of antennas.