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On the number of modes of a Gaussian mixture

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Original languageEnglish
Title of host publicationScale Space Methods in Computer Vision
Subtitle of host publication4th International Conference, Scale Space 2003 Isle of Skye, UK, June 10–12, 2003 Proceedings
PublisherSpringer Berlin Heidelberg
Pages625-640
Number of pages16
ISBN (Electronic)978-3-540-44935-5
ISBN (Print)978-3-540-40368-5
Publication statusPublished - 2003

Publication series

NameLecture Notes in Computer Science
Volume2695
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Abstract

We consider a problem intimately related to the creation of maxima under Gaussian blurring: the number of modes of a Gaussian mixture in D dimensions. To our knowledge, a general answer to this question is not known. We conjecture that if the components of the mixture have the same covariance matrix (or the same covariance matrix up to a scaling factor), then the number of modes cannot exceed the number of components. We demonstrate that the number of modes can exceed the number of components when the components are allowed to have arbitrary and different covariance matrices. We will review related results from scale-space theory, statistics and machine learning, including a proof of the conjecture in 1D. We present a convergent, EM-like algorithm for mode finding and compare results of searching for all modes starting from the centers of the mixture components with a brute-force search. We also discuss applications to data reconstruction and clustering.

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