On the Number of Modes of a Gaussian Mixture
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.
KeywordsConvex Hull Covariance Matrice Gaussian Kernel Multivalued Mapping Convex Linear Combination
Unable to display preview. Download preview PDF.
- 1.Carreira-Perpiñán, M.Á., Williams, C.K.I.: On the number of modes of a Gaussian mixture. Technical Report EDI-INF-RR-0159, School of Informatics, University of Edinburgh, UK (2003). Available online at http://www.informatics.ed.ac.uk/publications/report/0159.html.Google Scholar
- 2.Carreira-Perpiñán, M.Á.: Continuous Latent Variable Models for Dimensionality Reduction and Sequential Data Reconstruction. PhD thesis, Dept. of Computer Science, University of Sheffield, UK (2001)Google Scholar
- 3.Carreira-Perpiñán, M.Á.: Mode-finding for mixtures of Gaussian distributions. Technical Report CS-99-03, Dept. of Computer Science, University of Sheffield, UK (1999), revised August 4, 2000. Available online at http://www.dcs.shef.ac.uk/~miguel/papers/cs-99-03.html.Google Scholar
- 5.Lindeberg, T.: Scale-Space Theory in Computer Vision. Kluwer Academic Publishers Group, Dordrecht, The Netherlands (1994)Google Scholar
- 11.Kuijper, A., Florack, L.M.J.: The application of catastrophe theory to image analysis. Technical Report UU-CS-2001-23, Dept. of Computer Science, Utrecht University (2001). Available online at ftp://ftp.cs.uu.nl/pub/RUU/CS/techreps/CS-2001/2001-23.pdf.Google Scholar
- 12.Kuijper, A., Florack, L.M.J.: The relevance of non-generic events in scale space models. In Heyden, A., Sparr, G., Nielsen, M., Johansen, P., eds.: Proc. 7th European Conf. Computer Vision (ECCV’02), Copenhagen, Denmark (2002)Google Scholar
- 14.Silverman, B.W.: Using kernel density estimates to investigate multimodality. Journal of the Royal Statistical Society, B 43 (1981) 97–99Google Scholar
- 17.McLachlan, G.J., Krishnan, T.: The EM Algorithm and Extensions. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons (1997)Google Scholar
- 22.Bishop, C.M.: Neural Networks for Pattern Recognition. Oxford University Press, New York, Oxford (1995)Google Scholar
- 24.Carreira-Perpiñán, M.Á.: Reconstruction of sequential data with probabilistic models and continuity constraints. In Solla, S.A., Leen, T.K., Müller, K.R., eds.: Advances in Neural Information Processing Systems. Volume 12, MIT Press, Cambridge, MA (2000) 414–420Google Scholar