The in-sphere property for matrix means is studied. It is proved that the matrix power mean satisfies in-sphere property with respect to the Hilbert-Schmidt norm. A new characterization of the matrix arithmetic mean is provided. Some reverse AGM inequalities involving unitarily invariant norms and operator monotone functions are also obtained.

In this paper we introduce the weighted Hellinger distance for matrices which is an interpolating between the Euclidean distance and the Hellinger distance. We show the equivalence of the weighted Hellinger distance and the Alpha Procrustes distance. As a consequence, we prove that the matrix power mean μ p (t,A,B) = (tA p +(1 −t)B p) 1/p satisfies in-betweenness property in the weighted Hellinger and Alpha Procrustes distances.

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In this paper we propose a novel rough-fuzzy hybridization technique to feature selection and feature ranking problem. The idea is to model the local preference relation between pair of features by intuitionistic fuzzy values and search for a feature ranking that is consistent with those constraints. We apply the techniques used in group decision making where constraints are presented in form of intuitionistic fuzzy preference relation. The proposed method has been illustrated by some simple examples and verified on a benchmark dataset.