The speed at which millimeter-Wave (mmWave) channel estimation can be carried out is critical for the adoption of mmWave technologies. This is particularly crucial because mmWave transceivers are equipped with large antenna arrays to combat severe path losses, which consequently creates large channel matrices, whose estimation may incur significant overhead. This paper focuses on the mmWave channel estimation problem. Our objective is to reduce the number of measurements required to reliably estimate the channel. Specifically, channel estimation is posed as a "source compression" problem in which measurements mimic an encoded (compressed) version of the channel. Decoding the observed measurements, a task which is traditionally computationally intensive, is performed using a deep-learning-based approach, facilitating a high-performance channel discovery. Our solution not only outperforms state-of-the-art compressed sensing methods, but it also determines the lower bound on the number of measurements required for reliable channel discovery.