Representing distributions over permutations can be a daunting task due to the fact that the number of permutations of n objects scales factorially in n. One recent way that has been used to reduce storage complexity has been to exploit probabilistic independence, but as we argue, full independence assumptions impose strong sparsity constraints on distributions and are unsuitable for modeling rankings. We identify a novel class of independence structures, called riffled independence, which encompasses a more expressive family of distributions while retaining many of the properties necessary for performing efficient inference and reducing sample complexity. In riffled independence, one draws two permutations independently, then performs the riffle shuffle, common in card games, to combine the two permutations to form a single permutation. In ranking, riffled independence corresponds to ranking disjoint sets of objects independently, then interleaving those rankings. We provide a formal introduction and present algorithms for using riffled independence within Fourier-theoretic frameworks which have been explored by a number of recent papers. 1

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Modeling ranked data is an essential component in a number of important applications including recommendation systems and websearch. In many cases, judges omit preference among unobserved items and between unobserved and observed items. This case of analyzing incomplete rankings is very important from a practical perspective and yet has not been fully studied due to considerable computational difficulties. We show how to avoid such computational difficulties and efficiently construct a non-parametric model for rankings with missing items. We demonstrate our approach and show how it applies in the context of collaborative filtering. 1

Distributions over rankings are used to model data in various settings such as preference analysis and political elections. The factorial size of the space of rankings, however, typically forces one to make structural assumptions, such as smoothness, sparsity, or probabilistic independence about these underlying distributions. We approach the modeling problem from the computational principle that one should make structural assumptions which allow for efficient calculation of typical probabilistic queries. For ranking models, typical queries predominantly take the form of partial ranking queries (e.g., given a user's top-k favorite movies, what are his preferences over remaining movies?). In this paper, we argue that riffled independence factorizations proposed in recent literature [7, 8] are a natural structural assumption for ranking distributions, allowing for particularly efcient processing of partial ranking queries.

Representing distributions over permutations can be a daunting task due to the fact that the number of permutations of n objects scales factorially in n. One recent way that has been used to reduce storage complexity has been to exploit probabilistic independence, but as we argue, full independence assumptions impose strong sparsity constraints on distributions and are unsuitable for modeling rankings. We identify a novel class of independence structures, called riffled independence, encompassing a more expressive family of distributions while retaining many of the properties necessary for performing efficient inference and reducing sample complexity. In riffled independence, one draws two permutations independently, then performs the riffle shuffle, common in card games, to combine the two permutations to form a single permutation. In ranking, riffled independence corresponds to ranking disjoint sets of objects independently, then interleaving those rankings. We provide a formal introduction and present algorithms for using riffled independence within Fourier-theoretic frameworks which have been explored by a number of recent papers. 1

In typical ranking problems the total number n of items to be ranked is relatively large, but each data instance involves only k << n items. This paper examines the structure of such partial rankings in Fourier space. Specifically, we develop a kernel–based framework for solving ranking problems, define some canonical kernels on permutations, and show that by transforming to Fourier space, the complexity of computing the kernel between two partial rankings can be reduced from O((n−k)! 2) to O((2k) 2k+3). 1

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Abstract. We compare two recently proposed approaches for representing probability distributions over the space of permutations in the context of multi-target tracking. We show that these two representations, the Fourier approximation and the information form approximation can both be viewed as low dimensional projections of a true distribution, but with respect to different metrics. We identify the strengths and weaknesses of each approximation, and propose an algorithm for converting between the two forms, allowing for a hybrid approach that draws on the strengths of both representations. We show experimental evidence that there are situations where hybrid algorithms are favorable. 1

space algorithm for solving quadratic assignment problems

This paper presents a scalable algorithm for managing property information about moving objects tracked by a sensor network. Property information is obtained via distributed sensor observations, but will be corrupted when objects mix up with each other. The association between properties and objects then becomes ambiguous. We build a novel representation framework, exploiting an overcomplete Radon basis dictionary to model property uncertainty in such circumstances. By making use of the combinatorial structure of the basis design and sparse representations we can efficiently approximate the underlying probability distribution of the association between target properties and tracks, overcoming the exponential space that would otherwise be required. We conduct comparative simulations and the results validate the effectiveness of our approach.