Rees Matrix Constructions for Clustering of Data

Abstract

This paper continues the investigation of semigroup constructions motivated by applications in data mining. We give a complete description of the error-correcting capabilities of a large family of clusterers based on Rees matrix semigroups well known in semigroup theory. This result strengthens and complements previous formulas recently obtained in the literature. Examples show that our theorems do not generalize to other classes of semigroups.2000 Mathematics subject classification: primary 16S36, 20M35; secondary 20M25, 68T.

Optimization of Matrix Semirings for Classification Systems

“…We refer to [24,25,27] for a list of properties required of the sets of centroids. In particular, it is essential to find sets of centroids with large weights and small numbers of generators.…”

“…In particular, it is essential to find sets of centroids with large weights and small numbers of generators. The weight wt(v) of v ∈ F n is the number of nonzero components or coordinates in v. The weight of a set C ⊆ F n is the minimum weight of a nonzero element in C. For more information we refer the reader to [24,25].…”

“…We investigate properties of ideals in this construction motivated by applications to the design of centroid-based classification systems, or classifiers, as well as for multiple classifiers combining several initial classifiers (see the monograph [27], recent articles [24,25] and Section 2 below). The concept of an ideal is very important and has been considered in various branches of mathematics (see, for example [2] Optimization of matrix semirings 493 [1-3, 8, 9, 15, 16, 20, 23, 25, 26, 31-33]).…”

Optimal Rees Matrix Constructions for Analysis of Data

“…We refer to [16,20] for a list of properties required of the sets of centroids. In particular, it is essential to find sets of centroids with large weights and small numbers of generators.…”

On Hyperideals in Left Almost Semihypergroups

“…Ideal theory is important not only for the intrinsic interest and purity of its logical structure but because it is a necessary tool in many branches of mathematics and its applications such 2 ISRN Algebra as in informatics, physics, and others. As an example of applications of the concept of an ideal in informatics, let us mention that ideals of algebraic structures have been used recently to design efficient classification systems, see [8][9][10][11][12] .…”