On the Critical Points of Gaussian Mixtures
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This thesis is concerned with studying the question whether or not Gaussian mixtures have finitely many critical points. The relevance of this problem to the convergence of the mean-shift algorithm is discussed and an overview of some basic properties of the critical points of Gaussian mixtures is provided. Some previous results that are then reviewed include a reduction of this problem in the homoscedastic case and the construction of a very simple mixture with a large but finite number of critical points. A class of counterexamples is then presented that indicate that the inverse function theorem cannot be used to provide a direct solution to this problem. Finally, while the general problem is left unsolved, a proof is obtained in each of two special cases not previously seen in the literature.