Convexity of the inverse and Moore–Penrose inverse

In this article, we study the mean-squared-error performance of Bayesian direction-of-arrival (DOA) estimation in which prior belief about the target location is incorporated into the estimation process. Our primary result is an extension of the method of interval errors (MIE) to the case of maximum a posteriori (MAP) direction-of-arrival estimation. We work in a general framework in which the prior information used in the MAP estimation may not match the actual target distribution. In particular, when the prior is incorrect, the MAP estimator degrades relative to the performance of a MAP estimator with the correct prior. Our methods are able to accurately predict the performance of a MAP estimator in this more general situation. We apply our methods to investigate the sensitivity of MAP direction-of-arrival estimation to mismatches between the chosen prior and the actual angular distribution of the target.

Mean-Squared-Error Prediction for Bayesian Direction-of-Arrival Estimation

/pdf/power-matrix-means-and-related-inequalities-2p5c3iuemt.pdf

Power matrix means and related inequalities

Matrix convexity of the Moore-Penrose inverse was
 considered in the recent literature.
Here we give some converse inequalities as well as further generalizations.

/pdf/on-matrix-convexity-of-the-moore-penrose-inverse-17mpfxlodi.pdf

On matrix convexity of the Moore-Penrose inverse

The difficulty of establishing a (noncommutative) matrix inequality involving the geometric mean was discussed in 1978 by K.V. Bhagwat and R. Subramanian [9] who pointed out that the problem of defining a geometric mean for non-commutative operators “makes it difficult to establish the validity or otherwise of the classical inequalities involving the geometric mean”. However, in a recent paper, Sagae and Tanabe [32] define a geometric mean and establish an AG-GM inequality for a finite number of positive definite matrices.

The arithmetic mean — the geometric mean and related matrix inequalities

It is well-known that if A and B are two positive definite matrices of the same order and 0 ≤ λ ≤ 1, then . It is easy to construct an example consisting of two positive semi-definite matrices for which the above inequality is not true when one replaces the inverse operation by Moore-Penrose inverse operation. In this paper we give necessary and sufficient conditions for the validity of the inequality for every 0 ≤ λ ≤ 1. As an application, we give a sufficient condition under which the inequality (EA)+ λ E (A +) is valid, where A is a square matrix of random variables which is almost surely positive semi-definite, generalizing the well-known result (EA)− ≤ EA−1 when A is almost surely positive definite.

D. G. Kaffes

Papers

On the Matrix Convexity of the Moore-Penrose Inverse and Some Applications.