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Abstract:
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We consider the problem of density estimation under certain shape constraints. The notion of shape is quite flexible. It can mean a fixed number of modes in the density, say a bimodal or a trimodal function, or the number of modes plus a vector of function heights at the modes. The locations of these modes are left as variables, in order to fit to the data. The basic idea is to define a set of all valid densities (with the desired shape constraints) and to solve optimization problems (such as maximum likelihood estimation) on this set. This set is established using the "deformable template" theory -- choose a function from the correct class and use an appropriate action of the diffeomorphism group to form its orbits. Orbits define shape classes. The larger picture is to learn shape classes from the training data, and then to impose learnt shape constraints in estimating future functions from sparse, noisy data. We will present several examples of univariate and bivariate density estimation to illustrate our approach. (with Sutanoy Dasgupta, Ian Jermyn, and Debdeep Pati)
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