Alex Dimitrov's April 2011 Talk
Time and Date: 4:10 pm Thursday, April 7
Location: 5W Neill Hall, WSU Mathematics Department
Title: Soft Clustering Decoding of Neural Codes
Abstract: Methods based on Rate Distortion theory have been successfully used to cluster stimuli and neural responses in order to study neural codes at a level of detail supported by the amount of available data. They approximate the joint stimulus-response distribution by soft-clustering paired stimulus-response observations into smaller reproductions of the stimulus and response spaces. An optimal soft clustering is found by maximizing an information-theoretic cost function subject to both equality and inequality constraints, in hundreds to thousands of dimensions. The method of annealing has been used to solve the corresponding high dimensional non-linear optimization problems. The annealing solutions undergo a series of bifurcations in order to reach the optimum. We study that system using bifurcation theory in the presence of symmetries. The optimal models found by distortion methods have symmetries: any soft clustering data can lead to another equivalent model simply by permuting the labels of the classes. These symmetries are described by S_N, the algebraic group of all permutations on N symbols. The symmetry of the bifurcating solutions is dictated by the subgroup structure of S_N. In this presentation we describe these symmetry breaking bifurcations in detail, and indicate some of the consequences stemming from the form of the bifurcations.
Speaker Bio:
Background Information for Talk: Regarding reading, this was published a couple of years ago, link to paper here. The paper has too many technical details. For students that want to follow the talk but not get into all the nitty-gritty, the first 2 chapters of Golubitsky and Stewart, " The Symmetry Perspective: From equilibrium to chaos in phase space and physical space.", Birkhauser 2002 provide a good foundation for the topic. A more solid foundation would involve Guckenheimer and Holmes, "Nonlinear oscillations, dynamical systems and bifurcations of vector fileds", Ch 1-3 and Golubitsky and Schaeffer, "Singularities and Groups in Bifurcation Theory" V.II, Chapters XI - XIII. The topic falls under the theme "Equivariant bifurcation theory", if anybody wants to google it. Golubtisky and Stewart would be the best sources that I know of for original research and reviews.