This report provides a mathematically thorough review and investigation of Metric Multidimensional scaling (MDS) through the analysis of Euclidean distances in input and output spaces. By combining a geometric approach with modern linear algebra and multivariate analysis, Metric MDS is viewed as a Euclidean distance embedding transformation that converts between coordinate and coordinate-free r...

Practical statistical analysis of diffusion tensor images is considered, and we focus primarily on methods that use metrics based on Euclidean distances between powers of diffusion tensors. First we describe a family of anisotropy measures based on a scale invariant power-Euclidean metric, which are useful for visualisation. Some properties of the measures are derived and practical consideratio...

We give a metric characterization of snowflakes of Euclidean spaces. Namely, a metric space is isometric to Rn equipped with a distance (dE) , for some n ∈ N0 and ∈ (0, 1], where dE is the Euclidean distance, if and only if it is locally compact, 2-point isometrically homogeneous, and admits dilations of any factor.

We introduce a notion of the Euclidean and the Minkowski rank for arbitrary metric spaces and we study their behaviour with respect to products. We show that the Minkowski rank is additive with respect to metric products, while additivity of the Euclidean rank does not hold in general.

In solving pattern recognition problem in the Euclidean space, prototypes representing classes are de ned. On the other hand in the metric space, Nearest Neighbor method and K-Nearest Neighbor method are frequently used without de ning any prototypes. In this paper, we propose a new pattern recognition method for the metric space that can use prototypes which are the centroid of any three patte...

Let (X, d) be a metric space and m ∈ X. Suppose that φ : X×X → R is a nonnegative symmetric function. We define a metric d on X which is equivalent to d. If d is totally bounded, its completion is a compactification of (X, d). As examples, we construct two compactifications of (R, dE), where dE is the Euclidean metric and s ≥ 2. key words. equivalent metric; completion; compactification Mathema...

We investigate connections between pairs of (pseudo-)Riemannian metrics whose sum is a (tensor) product of a covector field with itself. A bijective mapping between the classes of Euclidean and Lorentzian metrics is constructed as a special result. The existence of such maps on a differentiable manifold is discussed. Similar relations for metrics of arbitrary signature on a manifold are conside...

Applying a theorem due to Belopol’ski and Birman, we show that the Laplace-Beltrami operator on 1-forms on R endowed with an asymptotically Euclidean metric has absolutely continuous spectrum equal to [0,+∞).

Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the Euclidean cluster diameter is large. Therefore, it is preferred to assign smaller costs to the paths that stay close to the input points. In this paper, we c...

k-Nearest Neighbours (k-NN) is a well understood and widely-used approach to classification and regression problems. In many cases, such applications of k-NN employ the standard Euclidean distance metric for the determination of the set of nearest neighbours to a particular test data sample. This paper investigates the use of a data-driven evolutionary approach, named KTree, for the automatic c...