Hierarchical GTM: constructing localized non-linear projection manifolds in a principled way

Peter Tino, Ian T. Nabney

Research output: Contribution to journalArticle

Abstract

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis ¸iteBishop98a in several directions: bf(1) We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping (GTM). bf(2) We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. bf(3) Using tools from differential geometry we derive expressions for local directional curvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the ancestor visualization plots which are captured by a child model. We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set and apply our system to two more complex 12- and 18-dimensional data sets.
Original languageEnglish
Pages (from-to)639-656
Number of pages18
JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
Volume24
Issue number5
DOIs
Publication statusPublished - May 2002

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Visualization
Projection
Probabilistic Model
Curvature
Differential Geometry
Hierarchical Model
EM Algorithm
Building Blocks
Regularization
Refinement
Model
Sufficient
Geometry
Formulation
Demonstrate
Statistical Models

Bibliographical note

©2002 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

Keywords

  • two-dimensional visualization plot
  • hierarchical visualization system
  • Generative Topographic Mapping (GTM)
  • hierarchical probabilistic models
  • ancestor visualization plots

Cite this

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Hierarchical GTM: constructing localized non-linear projection manifolds in a principled way. / Tino, Peter; Nabney, Ian T.

In: IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 24, No. 5, 05.2002, p. 639-656.

Research output: Contribution to journalArticle

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