1. An apparatus comprising: an equalizer to equalize a multidimensional signal transmitted over a communication channel and having a dimensionality of four or higher, the equalizer being adaptively trained based on an optimality criterion, wherein the multidimensional transmitted signal comprises at least one of polarization modulation, phase modulation, amplitude modulation, phase and amplitude modulation, phase and polarization modulation, amplitude and polarization modulation, or phase, amplitude and polarization modulation, wherein the equalizer produces soft outputs being used in soft channel decoding, and wherein the equalizer iteratively produces outputs using a priori information provided by a channel decoder.