A lot of machine learning algorithms, including clustering methods such as K-nearest neighbor (KNN), highly depend on the distance metrics to understand data pattern well and make right decision based data. In recent years, studies show that can significantly improve performance or deep model in clustering, classification, recovery tasks, etc. this article, we provide a survey widely used chall...

Given four complex matrices $A$‎, ‎$B$‎, ‎$C$ and $D$ where $Ainmathbb{C}^{ntimes n}$‎ ‎and $Dinmathbb{C}^{mtimes m}$ and let the matrix $left(begin{array}{cc}‎ A & B ‎ C & D‎ end{array} right)$ be a normal matrix and‎ assume that $lambda$ is a given complex number‎ ‎that is not eigenvalue of matrix $A$‎. ‎We present a method to calculate the distance norm (with respect to 2-norm) from $D$‎ to ...

A metric vector space is asymptotically metrically normable (AMN) if there exists a norm asymptotically isometric to the distance. We prove that AMN vector spaces are rigid in the class of metric vector spaces under asymptotically isometric perturbations. This result follows from a general metric normability criterium. If the distance is translation invariant and satisfies an approximate multip...

We show a continuity theorem for Stinespring’s dilation: two completely positive maps between arbitrary C∗-algebras are close in cb-norm iff we can find corresponding dilations that are close in operator norm. The proof establishes the equivalence of the cb-norm distance and the Bures distance for completely positive maps. We briefly discuss applications to quantum information theory.

Distance Education has become widespread in the United States. Many have questioned the appropriateness of distance education when the research clearly demonstrates that most learners prefer to learn in traditional educational settings. Because of the availability of sophisticated telecommunications systems there has been a redefinition of distance education, and an attempt to use technology to...

1 Distance metrics and similarity measures 2 1.1 Distance metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Vector norm and metric . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 The `p norm and `p metric . . . . . . . . . . . . . . . . . . . . . 4 1.4 Distance metric learning . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 The mean as a similarity measure . . . . . . ...