Thesis Proposal Probabilistic Reasoning with Permutations: A Fourier-Theoretic Approach

Permutations are ubiquitous in many real-world problems, such as voting, ranking, and data association. Representing uncertainty over permutations is challenging, since there are n! possibilities, and common factorized probability distribution representations, such as graphical models, are inefficient due to the mutual exclusivity constraints that are typically associated with permutations. This thesis explores a new approach for probabilistic reasoning with permutations based on the idea of approximating distributions using their low-frequency Fourier components. We use a generalized Fourier transform defined for functions on permutations, but unlike the widely used Fourier analysis on the circle or the real line, Fourier transforms of functions on permutations take the form of ordered collections of matrices. As we show, maintaining the appropriate set of low-frequency Fourier terms corresponds to maintaining matrices of simple marginal probabilities which summarize the underlying distribution. We show how to derive the Fourier coefficients of a variety of probabilistic models which arise in practice and that many useful models are either well-approximated or exactly represented by low-frequency (and in many cases, sparse) Fourier coefficient matrices. In addition to showing that Fourier representations are both compact and intuitive, we show how to cast common probabilistic inference operations in the Fourier domain, including marginalization, conditioning on evidence, and factoring based on probabilistic independence. The algorithms presented in this thesis are fully general and work gracefully in bandlimited settings where only a partial subset of Fourier coefficients is made available. From the theoretical side, we tackle several problems in understanding the consequences of the bandlimiting approximation. We present results in this thesis which illuminate the nature of error propagation in the Fourier domain and propose methods for mitigating their effects. Finally we demonstrate the effectiveness of our approach on several real datasets and show that our methods, in addition to being well-founded theoretically, are also scalable and provide superior results in practice.