On Communication through a Gaussian Channel with an MMSE Disturbance\n Constraint

This paper considers a Gaussian channel with one transmitter and two\nreceivers. The goal is to maximize the communication rate at the\nintended/primary receiver subject to a disturbance constraint at the\nunintended/secondary receiver. The disturbance is measured in terms of minimum\nmean square error (MMSE) of the interference that the transmission to the\nprimary receiver inflicts on the secondary receiver.\n The paper presents a new upper bound for the problem of maximizing the mutual\ninformation subject to an MMSE constraint. The new bound holds for vector\ninputs of any length and recovers a previously known limiting (when the length\nof vector input tends to infinity) expression from the work of Bustin\n$\\textit{et al.}$ The key technical novelty is a new upper bound on the MMSE.\nThis bound allows one to bound the MMSE for all signal-to-noise ratio (SNR)\nvalues $\\textit{below}$ a certain SNR at which the MMSE is known (which\ncorresponds to the disturbance constraint). This bound complements the\n`single-crossing point property' of the MMSE that upper bounds the MMSE for all\nSNR values $\\textit{above}$ a certain value at which the MMSE value is known.\nThe MMSE upper bound provides a refined characterization of the\nphase-transition phenomenon which manifests, in the limit as the length of the\nvector input goes to infinity, as a discontinuity of the MMSE for the problem\nat hand.\n For vector inputs of size $n=1$, a matching lower bound, to within an\nadditive gap of order $O \\left( \\log \\log \\frac{1}{\\sf MMSE} \\right)$ (where\n${\\sf MMSE}$ is the disturbance constraint), is shown by means of the mixed\ninputs technique recently introduced by Dytso $\\textit{et al.}$\n