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    Understanding High-Dimensional Spaces

    ISBN-10: 3642333974
    ISBN-13: 9783642333972
    Author(s): David B. Skillicorn
    Description: 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  More...
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    Publisher: Springer
    Binding: Paperback
    Pages: 108
    Size: 6.25" wide x 9.25" long x 0.25" tall
    Weight: 0.396
    Language: English

    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.

    Introduction
    A Natural Representation of Data Similarity
    Goals
    Outline
    Basic Structure of High-Dimensional Spaces
    Comparing Attributes
    Comparing Records
    Similarity
    High-Dimensional Spaces
    Summary
    Algorithms
    Improving the Natural Geometry
    Projection
    Singular Value Decompositions
    Random Projections
    Algorithms that Find Standalone Clusters
    Clusters Based on Density
    Parallel Coordinates
    Independent Component Analysis
    Latent Dirichlet Allocation
    Algorithms that Find Clusters and Their Relationships
    Clusters Based on Distance
    Clusters Based on Distribution
    Semidiscrete Decomposition
    Hierarchical Clustering
    Minimum Spanning Tree with Collapsing
    Overall Process for Constructing a Skeleton
    Algorithms that Wrap Clusters
    Distance-Based
    Distribution-Based
    1-Class Support Vector Machines
    Autoassociative Neural Networks
    Covers
    Algorithms to Place Boundaries Between Clusters
    Support Vector Machines
    Random Forests
    Overall Process for Constructing Empty Space
    Summary
    Spaces with a Single Center
    Using Distance
    Using Density
    Understanding the Skeleton
    Understanding Empty Space
    Summary
    Spaces with Multiple Centers
    What is a Cluster?
    Identifying Clusters
    Clusters Known Already
    Finding Clusters
    Finding the Skeleton
    Empty Space
    An Outer Boundary and Novel Data
    Interesting Data
    One-Cluster Boundaries
    One-Cluster-Against-the-Rest Boundaries
    Summary
    Representation by Graphs
    Building a Graph from Records
    Local Similarities
    Embedding Choices
    Using the Embedding for Clustering
    Summary
    Using Models of High-Dimensional Spaces
    Understanding Clusters
    Structure in the Set of Clusters
    Semantic Stratified Sampling
    Ranking Using the Skeleton
    Ranking Using Empty Space
    Applications to Streaming Data
    Concealment
    Summary
    Including Contextual Information
    What is Context?
    Changing Data
    Changing Analyst and Organizational Properties
    Changing Algorithmic Properties
    Letting Context Change the Models
    Recomputing the View
    Recomputing Derived Structures
    Recomputing the Clustering
    Summary
    Conclusions
    References
    Index

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