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18
Riffled Independence for Ranked Data
"... 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 ..."
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Cited by 9 (3 self)
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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 Fouriertheoretic frameworks which have been explored by a number of recent papers. 1
Efficient Probabilistic Inference with Partial Ranking Queries
"... 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 the ..."
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Cited by 3 (2 self)
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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 topk 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.
Estimating Probabilities in Recommendation Systems
"... 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 fr ..."
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Cited by 3 (3 self)
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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 nonparametric model for rankings with missing items. We demonstrate our approach and show how it applies in the context of collaborative filtering. 1
A FourierTheoretic Approach for Inferring Symmetries
"... In this paper, we propose a novel Fouriertheoretic approach for estimating the symmetry group G of a geometric object X. Our approach takes as input a geometric similarity matrix between loworder combinations of features of X and then searches within the tree of all feature permutations to detect ..."
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Cited by 1 (0 self)
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In this paper, we propose a novel Fouriertheoretic approach for estimating the symmetry group G of a geometric object X. Our approach takes as input a geometric similarity matrix between loworder combinations of features of X and then searches within the tree of all feature permutations to detect the sparse subset that defines the symmetry group G of X. Using the Fouriertheoretic approach, we construct an efficient marginalbased search strategy, which can recover the symmetry group G effectively. The framework introduced in this paper can be used to discover symmetries of more abstract geometric spaces and is robust to deformation noise. Experimental results show that our approach can fully determine the symmetries of many geometric objects. 1
Ranking with kernels in Fourier space
"... 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, de ..."
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Cited by 1 (0 self)
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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
Riffled Independence for Ranked Data (extended version with proofs)
"... 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 ..."
Abstract
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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 Fouriertheoretic frameworks which have been explored by a number of recent papers. 1
FourierInformation Duality in the Identity Management Problem
"... Abstract. We compare two recently proposed approaches for representing probability distributions over the space of permutations in the context of multitarget tracking. We show that these two representations, the Fourier approximation and the information form approximation can both be viewed as low ..."
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Abstract. We compare two recently proposed approaches for representing probability distributions over the space of permutations in the context of multitarget 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