Gradient of Mutual Information in Linear Vector Gaussian Channels in the Presence of Input Noise

This paper considers a general linear vector Gaussian channel with arbitrary signalling in the presence of Gaussian or Gaussian mixture input noise - i.e., noise added to a desired signal prior to its measurement. Generalising the fundamental relationship unveiled by Guo and extended by Palomar, we show for this scenario that the gradient of the mutual information between a desired signal - or its discrete class label - and a measured output with respect to the measurement matrix can be expressed in a novel form without a requirement for the approximations made in previous papers. We demonstrate that the derived expressions can outperform approximate gradient terms when integrated within a gradient ascent multi-objective optimisation approach.