Fast indexing in time sequence databases for similarity searching has attracted a lot of research recently. Most of the proposals, however, typically centered around the Euclidean distance and its derivatives. We examine the problem of multimodal similarity search in which users can choose the best one from multiple similarity models for their needs. In this paper, we present a novel and fast i...

In this paper we study number fields which are Euclidean with respect to a function different from the absolute value of the norm. We also show that the Euclidean minimum with respect to weighted norms may be irrational and not isolated.

Much of this section was copied and paraphrased from Heath’s Scientific Computing. Anyways. Suppose we are looking for an orthogonal transformation that annihilates desired components of a given vector. Recall that a square real matrix Q is said to be orthogonal if its columns are orthonormal, that is, that Q Q = I. Orthogonal transformations are nice because they preserve Euclidean norms of an...

We compute the number of summands in q-averages of norms needed to approximate an Euclidean norm. It turns out that these numbers depend on the norm involved essentially only through the maximal ratio of the norm and the Euclidean norm. Particular attention is given to the case q = ∞ (in which the average is replaced with the maxima). This is closely connected with the behavior of certain famil...

The isotropic elastic moduli closest to a given anisotropic elasticity tensor are defined using three definitions of elastic distance, the standard Frobenius (Euclidean) norm, the Riemannian distance for tensors, and the log-Euclidean norm. The closest moduli are unique for the Riemannian and the log-Euclidean norms, independent of whether the difference in stiffness or compliance is considered...

This paper introduces non-Euclidean c-means clustering algorithms. These algorithms rely on weighted norms to measure the distance between the feature vectors and the prototypes that represent the clusters. The proposed algorithms are developed by solving a constrained minimization problem in an iterative fashion. The norm weights are determined from the data in an attempt to produce partitions...