  This paper addresses the design of transmit precoder and receive combiner
matrices to support $N_{\rm s}$ independent data streams over a time-division
duplex (TDD) point-to-point massive multiple-input multiple-output (MIMO)
channel with either a fully digital or a hybrid structure. The optimal precoder
and combiner design amounts to finding the top-$N_{\rm s}$ singular vectors of
the channel matrix, but the explicit estimation of the entire high-dimensional
channel would require significant pilot overhead. Alternatively, prior works
suggest to find the precoding and combining matrices directly by exploiting
channel reciprocity and by using the power iteration method, but its
performance degrades in the low SNR regime. To tackle this challenging problem,
this paper proposes a learning-based active sensing framework, where the
transmitter and the receiver send pilots alternately using sensing beamformers
that are actively designed as functions of previously received pilots. This is
accomplished by using recurrent neural networks to summarize information from
the historical observations into hidden state vectors, then using fully
connected neural networks to learn the appropriate sensing beamformers in the
next pilot stage and finally the transmit precoding and receive combiner
matrices for data communications. Simulations demonstrate that the
learning-based method outperforms existing approaches significantly and
maintains superior performance even in the low SNR regime for both the fully
digital and hybrid MIMO scenarios.
