This paper develops theory and algorithms concerning a new metric for clustering data. The metric minimizes the total volume of clusters, where volume of a cluster is defined as the volume of the minimum volume ellipsoid (MVE) enclosing all data points in the cluster. This metric has the scale-invariant property, that is, the optimal clusters are invariant under an affine transformation of the ...

This paper develops theory and algorithms concerning a new metric for clustering data. The metric minimizes the total volume of clusters, where the volume of a cluster is defined as the volume of the minimum volume ellipsoid (MVE) enclosing all data points in the cluster. This metric is scale-invariant, that is, the optimal clusters are invariant under an affine transformation of the data space...

ABSTRACT There are various useful metrics for finding the distance between two points in Euclidean space. Metrics for finding the distance between two rigid body locations in Euclidean space depend on both the coordinate frame and units used. A metric independent of these choices is desirable. This paper presents a metric for a finite set of rigid body displacements. The methodology uses the pr...

We simplify the Hitchin's description of SU (2)-invariant self-dual Einstein metrics, making use of the tau-function of related four-pole Schlesinger system. The SU (2) invariant self-dual Einstein metrics were studied in a number of papers [1, 2, 3]. The local classification of the metrics of this type was given in extensive paper by Hitchin [3]. However, the final form of the metric coefficie...

On a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive probability densities, that is invariant under the action of the diffeomorphism group, is a multiple of the Fisher–Rao metric. Introduction. The Fisher–Rao metric on the space Prob(M) of probability densities is of importance in the field of information geometry. Restricted to...

We give a geometric derivation of SLE(κ, ρ) in terms of con-formally invariant random growing compact subsets of polygons. The parameters ρ j are related to the exterior angles of the polygons. We also show that SLE(κ, ρ) can be generated by a metric Brownian motion , where metric and Brownian motion are coupled and the metric is a pull-back metric of the Euclidean metric of an evolving polygon.

The Virasoro-Bott group endowed with the right-invariant L2metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.

In this work we study a particular class of Lie bialgebras arising from Hermitian structures on Lie algebras such that the metric is ad-invariant. We will refer to them as Lie bialgebras of complex type. These give rise to Poisson Lie groups G whose corresponding duals G∗ are complex Lie groups. We also prove that a Hermitian structure on g with ad-invariant metric induces a structure of the sa...

In this paper we study generalized weights as an algebraic invariant of a code. We first describe anticodes in the Hamming and in the rank metric, proving in particular that optimal anticodes in the rank metric coincide with Frobenius-closed spaces. Then we characterize both generalized Hamming and rank weights of a code in terms of the intersection of the code with optimal anticodes in the res...

The explicit complete Einstein-Kähler metric on the second type Cartan-Hartogs domain YII(r, p;K) is obtained in this paper when the parameter K equals p 2 + 1 p+1 . The estimate of holomorphic sectional curvature under this metric is also given which intervenes between −2K and − 2K p and it is a sharp estimate. In the meantime we also prove that the complete Einstein-Kähler metric is equivalen...