[PDF] Clustering Via Minimum Volume Ellipsoids eBook

Author : Mahesh Kumar
Publisher :
Page : 9 pages
File Size : 24,29 MB
Release : 2006
Category :
ISBN :

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This paper develops theory and algorithms concerning a new metric for clustering data. The metric minimizes the total volume of clusters, where volume of a cluster is defined as the volume of the minimum volume ellipsoid (MVE) enclosing all data points in the cluster. This metric has the scale-invariant property, that is, the optimal clusters are invariant under an affine transformation of the data space. We introduce the concept of outliers in the new metric and show that the proposed method of treating outliers asymptotically recovers the data distribution when the data comes from a single multivariate Gaussian distribution. Two heuristical algorithm are presented that attempt to optimize the new metric. On a series of empirical studies on real and simulated data sets, we show that volume-based clustering out-performs the k-means algorithm.

Author : Michael J. Todd
Publisher : SIAM
Page : 156 pages
File Size : 18,30 MB
Release : 2016-07-11
Category : Mathematics
ISBN : 1611974380

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This book, the first on these topics, addresses the problem of finding an ellipsoid to represent a large set of points in high-dimensional space, which has applications in computational geometry, data representations, and optimal design in statistics. The book covers the formulation of this and related problems, theoretical properties of their optimal solutions, and algorithms for their solution. Due to the high dimensionality of these problems, first-order methods that require minimal computational work at each iteration are attractive. While algorithms of this kind have been discovered and rediscovered over the past fifty years, their computational complexities and convergence rates have only recently been investigated. The optimization problems in the book have the entries of a symmetric matrix as their variables, so the author's treatment also gives an introduction to recent work in matrix optimization. This book provides historical perspective on the problems studied by optimizers, statisticians, and geometric functional analysts; demonstrates the huge computational savings possible by exploiting simple updates for the determinant and the inverse after a rank-one update, and highlights the difficulties in algorithms when related problems are studied that do not allow simple updates at each iteration; and gives rigorous analyses of the proposed algorithms, MATLAB codes, and computational results.

Author : Slawomir Wierzchoń
Publisher : Springer
Page : 433 pages
File Size : 39,67 MB
Release : 2017-12-29
Category : Technology & Engineering
ISBN : 3319693085

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This book provides the reader with a basic understanding of the formal concepts of the cluster, clustering, partition, cluster analysis etc. The book explains feature-based, graph-based and spectral clustering methods and discusses their formal similarities and differences. Understanding the related formal concepts is particularly vital in the epoch of Big Data; due to the volume and characteristics of the data, it is no longer feasible to predominantly rely on merely viewing the data when facing a clustering problem. Usually clustering involves choosing similar objects and grouping them together. To facilitate the choice of similarity measures for complex and big data, various measures of object similarity, based on quantitative (like numerical measurement results) and qualitative features (like text), as well as combinations of the two, are described, as well as graph-based similarity measures for (hyper) linked objects and measures for multilayered graphs. Numerous variants demonstrating how such similarity measures can be exploited when defining clustering cost functions are also presented. In addition, the book provides an overview of approaches to handling large collections of objects in a reasonable time. In particular, it addresses grid-based methods, sampling methods, parallelization via Map-Reduce, usage of tree-structures, random projections and various heuristic approaches, especially those used for community detection.

Author : Panos M. Pardalos
Publisher : American Mathematical Soc.
Page : 252 pages
File Size : 15,16 MB
Release : 2008-04-09
Category : Computers
ISBN : 9780821870402

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Data mining aims at finding interesting, useful or profitable information in very large databases. The enormous increase in the size of available scientific and commercial databases (data avalanche) as well as the continuing and exponential growth in performance of present day computers make data mining a very active field. In many cases, the burgeoning volume of data sets has grown so large that it threatens to overwhelm rather than enlighten scientists. Therefore, traditional methods are revised and streamlined, complemented by many new methods to address challenging new problems. Mathematical Programming plays a key role in this endeavor. It helps us to formulate precise objectives (e.g., a clustering criterion or a measure of discrimination) as well as the constraints imposed on the solution (e.g., find a partition, a covering or a hierarchy in clustering). It also provides powerful mathematical tools to build highly performing exact or approximate algorithms. This book is based on lectures presented at the workshop on "Data Mining and Mathematical Programming" (October 10-13, 2006, Montreal) and will be a valuable scientific source of information to faculty, students, and researchers in optimization, data analysis and data mining, as well as people working in computer science, engineering and applied mathematics.

Author : Petra PErner
Publisher : Springer
Page : 340 pages
File Size : 43,48 MB
Release : 2011-08-09
Category : Computers
ISBN : 3642231845

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This book constitutes the refereed proceedings of the 11th Industrial Conference on Data Mining, ICDM 2011, held in New York, USA in September 2011. The 22 revised full papers presented were carefully reviewed and selected from 100 submissions. The papers are organized in topical sections on data mining in medicine and agriculture, data mining in marketing, data mining for Industrial processes and in telecommunication, Multimedia Data Mining, theoretical aspects of data mining, Data Warehousing, WebMining and Information Mining.

Author : Peng Sun
Publisher :
Page : 0 pages
File Size : 38,13 MB
Release : 2002
Category :
ISBN :

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We present a practical algorithm for computing the minimum volume n-dimensional ellipsoid that must contain m given points a_1, ..., a_m in R^n. This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interior-point methods, and it has been the subject of much theoretical complexity analysis. Here we focus on computation. We present a combined interior-point and active-set method for solving this problem. Our computational results demonstrate that our method solves very large problem instances (m=30,000 and n=30) to a high degree of accuracy in under 30 seconds on a personal computer.

Author : Christian Hennig
Publisher : CRC Press
Page : 753 pages
File Size : 23,80 MB
Release : 2015-12-16
Category : Business & Economics
ISBN : 1466551895

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Handbook of Cluster Analysis provides a comprehensive and unified account of the main research developments in cluster analysis. Written by active, distinguished researchers in this area, the book helps readers make informed choices of the most suitable clustering approach for their problem and make better use of existing cluster analysis tools.The

Author : Jacob E. Goodman
Publisher : Cambridge University Press
Page : 640 pages
File Size : 28,7 MB
Release : 2005-08-08
Category : Computers
ISBN : 9780521848626

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This 2005 book deals with interest topics in Discrete and Algorithmic aspects of Geometry.

Author : Sorin Draghici
Publisher : CRC Press
Page : 1036 pages
File Size : 23,14 MB
Release : 2016-04-19
Category : Computers
ISBN : 1439809763

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Richly illustrated in color, Statistics and Data Analysis for Microarrays Using R and Bioconductor, Second Edition provides a clear and rigorous description of powerful analysis techniques and algorithms for mining and interpreting biological information. Omitting tedious details, heavy formalisms, and cryptic notations, the text takes a hands-on,