Understanding High-Dimensional Spaces

by Skillicorn, David B.

• ISBN13: 9783642333972
• ISBN10: 3642333974
• Format: Paperback
• Copyright: 2012-09-27
• Publisher: Springer-Nature New York Inc

Summary

High-dimensional spaces arise as a way of modelling datasets with many attributes. Such a dataset can be directly represented in a space spanned by its attributes, with each record represented as a point in the space with its position depending on its attribute values. Such spaces are not easy to work with because of their high dimensionality: our intuition about space is not reliable, and measures such as distance do not provide as clear information as we might expect. There are three main areas where complex high dimensionality and large datasets arise naturally: data collected by online retailers, preference sites, and social media sites, and customer relationship databases, where there are large but sparse records available for each individual; data derived from text and speech, where the attributes are words and so the corresponding datasets are wide, and sparse; and data collected for security, defense, law enforcement, and intelligence purposes, where the datasets are large and wide. Such datasets are usually understood either by finding the set of clusters they contain or by looking for the outliers, but these strategies conceal subtleties that are often ignored. In this book the author suggests new ways of thinking about high-dimensional spaces using two models: a skeleton that relates the clusters to one another; and boundaries in the empty space between clusters that provide new perspectives on outliers and on outlying regions. The book will be of value to practitioners, graduate students and researchers.

Table of Contents

Introduction	p. 1
A Natural Representation of Data Similarity	p. 3
Goals	p. 8
Outline	p. 10
Basic Structure of High-Dimensional Spaces	p. 13
Comparing Attributes	p. 13
Comparing Records	p. 14
Similarity	p. 14
High-Dimensional Spaces	p. 16
Summary	p. 18
Algorithms	p. 19
Improving the Natural Geometry	p. 19
Projection	p. 20
Singular Value Decompositions	p. 20
Random Projections	p. 22
Algorithms that Find Standalone Clusters	p. 23
Clusters Based on Density	p. 23
Parallel Coordinates	p. 24
Independent Component Analysis	p. 24
Latent Dirichlet Allocation	p. 25
Algorithms that Find Clusters and Their Relationships	p. 25
Clusters Based on Distance	p. 25
Clusters Based on Distribution	p. 26
Semidiscrete Decomposition	p. 27
Hierarchical Clustering	p. 29
Minimum Spanning Tree with Collapsing	p. 29
Overall Process for Constructing a Skeleton	p. 30
Algorithms that Wrap Clusters	p. 31
Distance-Based	p. 32
Distribution-Based	p. 32
1-Class Support Vector Machines	p. 32
Autoassociative Neural Networks	p. 33
Covers	p. 34
Algorithms to Place Boundaries Between Clusters	p. 34
Support Vector Machines	p. 35
Random Forests	p. 35
Overall Process for Constructing Empty Space	p. 36
Summary	p. 37
Spaces with a Single Center	p. 39
Using Distance	p. 39
Using Density	p. 40
Understanding the Skeleton	p. 42
Understanding Empty Space	p. 43
Summary	p. 45
Spaces with Multiple Centers	p. 47
What is a Cluster?	p. 48
Identifying Clusters	p. 50
Clusters Known Already	p. 50
Finding Clusters	p. 50
Finding the Skeleton	p. 55
Empty Space	p. 58
An Outer Boundary and Novel Data	p. 58
Interesting Data	p. 60
One-Cluster Boundaries	p. 63
One-Cluster-Against-the-Rest Boundaries	p. 63
Summary	p. 64
Representation by Graphs	p. 67
Building a Graph from Records	p. 68
Local Similarities	p. 68
Embedding Choices	p. 69
Using the Embedding for Clustering	p. 70
Summary	p. 71
Using Models of High-Dimensional Spaces	p. 73
Understanding Clusters	p. 73
Structure in the Set of Clusters	p. 76
Semantic Stratified Sampling	p. 77
Ranking Using the Skeleton	p. 78
Ranking Using Empty Space	p. 87
Applications to Streaming Data	p. 89
Concealment	p. 90
Summary	p. 91
Including Contextual Information	p. 93
What is Context?	p. 93
Changing Data	p. 93
Changing Analyst and Organizational Properties	p. 94
Changing Algorithmic Properties	p. 95
Letting Context Change the Models	p. 95
Recomputing the View	p. 95
Recomputing Derived Structures	p. 96
Recomputing the Clustering	p. 97
Summary	p. 98
Conclusions	p. 99
References	p. 103
Index	p. 107
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