TR on the number of modes of a Gaussian mixture

[Miguel . Carreira-Perpin]

We would like to announce a new TR:

  An isotropic Gaussian mixture can have more modes than components

  Miguel Carreira-Perpinan and Chris Williams

Available from

  http://www.inf.ed.ac.uk/publications/report/0185.html


It is well known that in 1-d a mixture of isotropic Gaussians can have
no more modes than components (see e.g. Silverman (1981), Yuille and
Poggio (1986)). In this TR we show that in d dimensions (with d >= 2)
a mixture of M > 2 isotropic Gaussians can have more than M modes. We
first discuss a 3-component mixture in d = 2 where the Gaussians are
located at the vertices of an equilateral triangle. For a certain
range of variances modes are present near to the vertices and also at
the centre of the triangle. (The equilateral triangle construction was
suggested by Prof. J. J. Duistermaat, personal communication, 2003.)
We also extend the construction to the regular simplex with M vertices
and show that for M > 2 there is always a range of variances for which
M+1 modes are present.

Miguel Carreira-Perpinan
Chris Williams

-- 
Miguel A Carreira-Perpinan
Dept. of Computer Science, Rm 283      Tel. (416) 9463986
University of Toronto                  Fax  (416) 9781455
6 King's College Road                  mailto:miguel at cs.toronto.edu
Toronto, ON M5S 3H5, Canada            http://www.cs.toronto.edu/~miguel