  Millimeter wave (mmWave) massive multiple-input multiple-output (MIMO)
systems typically employ hybrid mixed signal processing to avoid expensive
hardware and high training overheads. {However, the lack of fully digital
beamforming at mmWave bands imposes additional challenges in channel
estimation. Prior art on hybrid architectures has mainly focused on greedy
optimization algorithms to estimate frequency-flat narrowband mmWave channels,
despite the fact that in practice, the large bandwidth associated with mmWave
channels results in frequency-selective channels. In this paper, we consider a
frequency-selective wideband mmWave system and propose two deep learning (DL)
compressive sensing (CS) based algorithms for channel estimation.} The proposed
algorithms learn critical apriori information from training data to provide
highly accurate channel estimates with low training overhead. In the first
approach, a DL-CS based algorithm simultaneously estimates the channel supports
in the frequency domain, which are then used for channel reconstruction. The
second approach exploits the estimated supports to apply a low-complexity
multi-resolution fine-tuning method to further enhance the estimation
performance. Simulation results demonstrate that the proposed DL-based schemes
significantly outperform conventional orthogonal matching pursuit (OMP)
techniques in terms of the normalized mean-squared error (NMSE), computational
complexity, and spectral efficiency, particularly in the low signal-to-noise
ratio regime. When compared to OMP approaches that achieve an NMSE gap of
\$\unit[\{4-10\}]{dB}\$ with respect to the Cramer Rao Lower Bound (CRLB), the
proposed algorithms reduce the CRLB gap to only \$\unit[\{1-1.5\}]{dB}\$, while
significantly reducing complexity by two orders of magnitude.
