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Abstract:
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We consider the problem of nonparametric regression. One approach is to represent the function in a series expansion using a linear combination of basis functions. Overcomplete dictionaries provide a redundant collection of generating elements, however, coefficients in the expansion are no longer unique. Despite the non-uniqueness, this has the potential to lead to sparser representations by using fewer non-zero coefficients. We show that Levy random fields are ideally suited for construction of priors on functions using these overcomplete representations for the general nonparametric regression problem, and provide a natural limiting generalization of priors for the finite dimensional version of the regression problem. We compare this approach to other penalized likelihood methods, such as the LASSO, and discuss methods for posterior inference.
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