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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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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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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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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
Efficient Sampling for Bipartite Matching Problems
"... Bipartite matching problems characterize many situations, ranging from ranking in information retrieval to correspondence in vision. Exact inference in realworld applications of these problems is intractable, making efficient approximation methods essential for learning and inference. In this paper ..."
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Bipartite matching problems characterize many situations, ranging from ranking in information retrieval to correspondence in vision. Exact inference in realworld applications of these problems is intractable, making efficient approximation methods essential for learning and inference. In this paper we propose a novel sequential matching sampler based on a generalization of the PlackettLuce model, which can effectively make large moves in the space of matchings. This allows the sampler to match the difficult target distributions common in these problems: highly multimodal distributions with well separated modes. We present experimental results with bipartite matching problems—ranking and image correspondence—which show that the sequential matching sampler efficiently approximates the target distribution, significantly outperforming other sampling approaches. 1
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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 proble ..."
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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 Efficient Inference with Partial Rankings
"... Distributions over rankings are used to model data in a multitude of real world settings such as preference analysis and political elections. Modeling such distributions presents several computational challenges, however, due to the factorial size of the set of rankings over an item set. Some of the ..."
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Distributions over rankings are used to model data in a multitude of real world settings such as preference analysis and political elections. Modeling such distributions presents several computational challenges, however, due to the factorial size of the set of rankings over an item set. Some of these challenges are quite familiar to the artificial intelligence community, such as how to compactly represent a distribution over a combinatorially large space, and how to efficiently perform probabilistic inference with these representations. With respect to ranking, however, there is the additional challenge of what we refer to as human task complexity — users are rarely willing to provide a full ranking over a long list of candidates, instead often preferring to provide partial ranking information. Simultaneously addressing all of these challenges — i.e., designing a compactly representable model which is amenable to efficient inference and can be learned using partial ranking data — is a difficult task, but is necessary if we would like to scale to problems with nontrivial size. In this paper, we show that the recently proposed riffled independence assumptions cleanly and efficiently address each of the above challenges. In particular, we
Probabilistic Models over Ordered Partitions with Applications in Document Ranking and Collaborative Filtering
"... Ranking is an important task for handling a large amount of content. Ideally, training data for supervised ranking would include a complete rank of documents (or other objects such as images or videos) for a particular query. However, this is only possible for small sets of documents. In practice, o ..."
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Ranking is an important task for handling a large amount of content. Ideally, training data for supervised ranking would include a complete rank of documents (or other objects such as images or videos) for a particular query. However, this is only possible for small sets of documents. In practice, one often resorts to document rating, in that a subset of documents is assigned with a small number indicating the degree of relevance. This poses a general problem of modelling and learning rank data with ties. In this paper, we propose a probabilistic generative model, that models the process as permutations over partitions. This results in superexponential combinatorial state space with unknown numbers of partitions and unknown ordering among them. We approach the problem from the discrete choice theory,