We present polynomial upper and lower bounds on the number of iterations performed by Lloyd’s method for k-means clustering. Our upper bounds are polynomial in the number of points, number of clusters, and the spread of the point set. We also present a lower bound, showing that in the worst case the k-means heuristic needs to perform Ω(n) iterations, for n points on the real line and two center...

k-means++ is a seeding technique for the k-means method with an expected approximation ratio of O(log k), where k denotes the number of clusters. Examples are known on which the expected approximation ratio of k-means++ is Ω(log k), showing that the upper bound is asymptotically tight. However, it remained open whether k-means++ yields an O(1)-approximation with probability 1/poly(k) or even wi...

We investigate, theoretically and empirically, the effectiveness of kernel K-means++ samples as landmarks in the Nyström method for low-rank approximation of kernel matrices. Previous empirical studies (Zhang et al., 2008; Kumar et al., 2012) observe that the landmarks obtained using (kernel) K-means clustering define a good lowrank approximation of kernel matrices. However, the existing work d...

We propose a method for creating different types of study groups with aim to support effective collaboration during learning. We concentrate on the small groups which solve short-term well-defined problems. The method is able to apply many types of students’ characteristics as inputs, e.g. interests, knowledge, but also their collaborative characteristics. It is based on the Group Technology ap...

<span>Learning class is a collection of several students in an educational institution. Every beginning the school year institution conducts grouping test. However, sometimes not accordance with ability students. For this reason, system needed to be able see according desired parameters. Determination weight test scores done using K-Means method as method. Iteration or repetition process ...

Organizing data into semantically more meaningful is one of the fundamental modes of understanding and learning. Cluster analysis is a formal study of methods for understanding and algorithm for learning. K-mean clustering algorithm is one of the most fundamental and simple clustering algorithms. When there is no prior knowledge about the distribution of data sets, K-mean is the first choice fo...

Thirdand fourth-order accurate finite difference schemes for the first derivative of the square of the speed are developed, for both uniform and non-uniform grids, and applied in the study of a two-dimensional viscous fluid flow through an irregular domain. The von Mises transformation is used to transform the governing equations, and map the irregular domain onto a rectangular computational do...