In information retrieval, documents(or images) are represented as vectors and the whole repository is represented as a matrix. Some similarity, distance and norm measurements between documents(or images) involve matrix computation. The challenge is the matrix may be too large to store, and compute. The idea is to reduce the matrix size while at the same time preserves characteristics such as Eu...

Locality preserving projection (LPP) is an effective dimensionality reduction method based on manifold learning, which is defined over the graph weighted squared 2-norm distances in the projected subspace. Since squared 2-norm distance is prone to outliers, it is desirable to develop a robust LPP method. In this paper, motivated by existing studies that improve the robustness of statistical lea...

In this paper, the connection between Menger probabilistic norms and H"{o}hle probabilistic norms is discussed. In addition, the correspondence between probabilistic norms and Wu-Fang fuzzy (semi-) norms is established. It is shown that a probabilistic norm (with triangular norm $min$) can generate a Wu-Fang fuzzy semi-norm and conversely, a Wu-Fang fuzzy norm can generate a probabilistic norm.

Data analysis and interpretation require intellectual frameworks, and to share the results in a community, scientific bases are preferred for the framework. A variety of indexes have been used for interpreting distances between transcriptome data, however, such indexes always lack necessity since they have been derived for non-biological purposes and their appropriateness have not been verified...

This paper presents the generalization of weighted distances to modules and their computation through the chamfer algorithm on general point-lattices. A first part is devoted to the definitions and properties (distance, metric, norm) of weighted distances on modules, with a presentation of weight optimization in the general case, to get rotation invariant distances. A general formula for the we...

The distance to uncontrollability for a linear control system is the distance (in the 2-norm) to the nearest uncontrollable system. We present an algorithm based on methods of Gu and Burke–Lewis–Overton that estimates the distance to uncontrollability to any prescribed accuracy. The new method requires O(n4) operations on average, which is an improvement over previous methods which have complex...

This paper presents a novel distance concept, Vector-Distance (VD) for high dimensional data. VD extends traditional scalar-distance to a vector-like fashion by collecting multi partial distances from diverse angles. These partial distances are derived from random projections, and they preserve individual features of dimensions as much as possible. Based on VD definition, a method family for ne...

In our paper we approximate a set of given points by a general circle. More precisely, we consider the problem of locating and scaling the unit ball of some given norm k1 with respect to fixed points on the plane such that the sum of weighted distances between the circle and the fixed points is minimized, where the distance is measured by a norm k2. We present results for the general case. In t...