Positive Definite Norm Dependent Matrices In Stochastic Modeling

A Note on Positive Definite Norm Dependent Functions

Let K be an origin symmetric star body in R. We prove, under very mild conditions on the function f : [0,∞) → R, that if the function f(‖x‖K) is positive definite on R , then the space (R, ‖ · ‖K) embeds isometrically in L0. This generalizes the solution to Schoenberg’s problem and leads to progress in characterization of n-dimensional versions, i.e. random vectors X = (X1, ...,Xn) in R n such ...

Definite and Quasidefinite Sets of Stochastic Matrices'

ON f-CONNECTIONS OF POSITIVE DEFINITE MATRICES

In this paper, by using Mond-Pečarić method we provide some inequalities for connections of positive definite matrices. Next, we discuss specifications of the obtained results for some special cases. In doing so, we use α-arithmetic, α-geometric and α-harmonic operator means.

Riemannian Sparse Coding for Positive Definite Matrices

Inspired by the great success of sparse coding for vector valued data, our goal is to represent symmetric positive definite (SPD) data matrices as sparse linear combinations of atoms from a dictionary, where each atom itself is an SPD matrix. Since SPD matrices follow a non-Euclidean (in fact a Riemannian) geometry, existing sparse coding techniques for Euclidean data cannot be directly extende...

Determinantal inequalities for positive definite matrices

Let Ai , i = 1, . . . ,m , be positive definite matrices with diagonal blocks A ( j) i , 16 j 6 k , where A ( j) 1 , . . . ,A ( j) m are of the same size for each j . We prove the inequality det( m ∑ i=1 A−1 i ) > det( m ∑ i=1 (A (1) i ) −1) · · ·det( m ∑ i=1 (A (k) i ) −1) and more determinantal inequalities related to positive definite matrices.