In his classical argument, Rao derives the Riemannian distance corresponding to Fisher metric using a mapping between space of positive measures and Euclidean space. He obtains Hellinger on full subset probability measures. order highlight interplay theory quantum information theory, we extend this construction positive-definite Hermitian matrices submersions quotient manifolds. The analog turn...

Geodesic distance matrices can reveal shape properties that are largely invariant to non-rigid deformations, and thus are often used to analyze and represent 3-D shapes. However, these matrices grow quadratically with the number of points. Thus for large point sets it is common to use a low-rank approximation to the distance matrix, which fits in memory and can be efficiently analyzed using met...

The reconstruction of phylogenetic relationships in the primate fossil record is dependent upon a thorough understanding of the phylogenetic utility of craniodental characters. Here, we test three previously proposed hypotheses for the propensity of primate craniomandibular data to exhibit homoplasy, using a study design based on the relative congruence between cranial distance matrices and a c...

Symmetric positive definite (spd) matrices pervade numerous scientific disciplines, including machine learning and optimization. We consider the key task of measuring distances between two spd matrices; a task that is often nontrivial whenever the distance function must respect the non-Euclidean geometry of spd matrices. Typical non-Euclidean distance measures such as the Riemannian metric δR(X...

Symmetric positive definite (spd) matrices pervade numerous scientific disciplines, including machine learning and optimization. We consider the key task of measuring distances between two spd matrices; a task that is often nontrivial whenever the distance function must respect the non-Euclidean geometry of spd matrices. Typical non-Euclidean distance measures such as the Riemannian metric δR(X...

This paper presents a new technique for computing the barycenter of a set of distance or kernel matrices. These matrices, which define the interrelationships between points sampled from individual domains, are not required to have the same size or to be in row-by-row correspondence. We compare these matrices using the softassign criterion, which measures the minimum distortion induced by a prob...

One of the aims of this paper is to solve an open problem of Lovász about relations between graph spectra and cut-distance. The paper starts with several inequalities between two versions of the cut-norm and the two largest singular values of arbitrary complex matrices, extending, in particular, the well-known graph-theoretical Expander Mixing Lemma and giving a hitherto unknown converse of it....

The stability radius of an n×n matrix A (or distance to instability) is a well-known measure of robustness of stability of the linear stable dynamical system ẋ = Ax. Such a distance is commonly measured either in the 2-norm or in the Frobenius norm. Even if the matrix A is real, the distance to instability is most often considered with respect to complex valued matrices (in such case the two no...

The celebrated Feng-Rao bound estimates the minimum distance of codes defined by means of their parity check matrices. From the Feng-Rao bound it is clear how to improve a large family of codes by leaving out certain rows in their parity check matrices. In this paper we derive a simple lower bound on the minimum distance of codes defined by means of their generator matrices. From our bound it i...

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...