We study a nonlinear vector precoding scheme which inverts the
wireless MIMO channel at the transmitter so that simple
symbol-by-symbol detection can be used at the receiver. In particular,
the transmit energy is minimized by relaxing the transmitted symbols
to a larger alphabet for precoding, which preserves the minimum
signaling distance. First, the average energy savings with random MIMO
channels is studied in the large-system limit using the replica
method, a technique invented in statistical physics. It is found that
significant gains can be achieved with complex-valued
alphabets. Secondly, we introduce polynomial-complexity precoding
schemes for BPSK and QPSK by using convex rather than discrete relaxed
alphabets. In case the number of transmit antennas is more than twice
the number of receive antennas, the new scheme, despite its polynomial
complexity, asymptotically outperforms NP-hard precoding using the
popular Tomlinson-Harashima signaling.