On the Minimum Mean $p$-th Error in Gaussian Noise Channels and its\n Applications

The problem of estimating an arbitrary random vector from its observation\ncorrupted by additive white Gaussian noise, where the cost function is taken to\nbe the Minimum Mean $p$-th Error (MMPE), is considered. The classical Minimum\nMean Square Error (MMSE) is a special case of the MMPE. Several bounds,\nproperties and applications of the MMPE are derived and discussed. The optimal\nMMPE estimator is found for Gaussian and binary input distributions. Properties\nof the MMPE as a function of the input distribution, SNR and order $p$ are\nderived. In particular, it is shown that the MMPE is a continuous function of\n$p$ and SNR. These results are possible in view of interpolation and change of\nmeasure bounds on the MMPE. The `Single-Crossing-Point Property' (SCPP) that\nbounds the MMSE for all SNR values {\\it above} a certain value, at which the\nMMSE is known, together with the I-MMSE relationship is a powerful tool in\nderiving converse proofs in information theory. By studying the notion of\nconditional MMPE, a unifying proof (i.e., for any $p$) of the SCPP is shown. A\ncomplementary bound to the SCPP is then shown, which bounds the MMPE for all\nSNR values {\\it below} a certain value, at which the MMPE is known. As a first\napplication of the MMPE, a bound on the conditional differential entropy in\nterms of the MMPE is provided, which then yields a generalization of the\nOzarow-Wyner lower bound on the mutual information achieved by a discrete input\non a Gaussian noise channel. As a second application, the MMPE is shown to\nimprove on previous characterizations of the phase transition phenomenon that\nmanifests, in the limit as the length of the capacity achieving code goes to\ninfinity, as a discontinuity of the MMSE as a function of SNR. As a final\napplication, the MMPE is used to show bounds on the second derivative of mutual\ninformation, that tighten previously known bounds.\n