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Low Distortion Embedding of the Hamming Space into a Sphere with Quadrance Metric and k-means Clustering of Nominal-continuous Data

Denisiuk A., Grabowski M.

Fundamenta Informaticae

2017

Vol. 153, nr 3

221--233

In this article we propose a new clustering algorithm for combinations of continuous and nominal data. The proposed algorithm is based on embedding of the nominal data into the unit sphere with a quadrance metrics, and adaptation of the general k-means clustering algorithm for the embedding data. It is also shown that the distortion of new embedding with respect to the Hamming metrics is less than that of other considered possibilities. A series of numerical experiments on real and synthetic datasets show that the proposed algorithm provide a comparable alternative to other clustering algorithms for combinations of continuous and nominal data.

Embedding theorems for holomorphic functions of several complex variables

Długosz R

Journal of Applied Analysis

2013

Vol. 19, nr 1

153--165

Many authors studied families XG of complex valued functions, which are holomorphic in bounded complete n-circular domains G⊂Cn and fulfill some geometric conditions. The above functions were applied later to research families of locally biholomorphic mappings in Cn. In this paper we consider a problem of inclusions between a few of such families XG and families MKG, k=2, 3, …, which are defined by applying a function decomposition with respect to the group of kth roots of unity.

En embedding theorem for unbounded convex sets in a Banach space

Bielawski J., Tabor J.

Demonstratio Mathematica

2009

Vol. 42, nr 4

703-709

Let V be a closed convex cone and Cv be a space of nonempty closed convex subsets of a Banach space such that their Hausdorff distance form the cone V is finite. In this paper we embed the space Cv isometrically and isomorphically into a Banach space.