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70
Analysis of systematic scan Metropolis algorithms using Iwahori–Hecke algebra techniques
 Michigan Math. J
, 2000
"... Abstract. We give the first analysis of a systematic scan version of the Metropolis algorithm. Our examples include generating random elements of a Coxeter group with probability determined by the length function. The analysis is based on interpreting Metropolis walks in terms of the multiplication ..."
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Cited by 34 (9 self)
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Abstract. We give the first analysis of a systematic scan version of the Metropolis algorithm. Our examples include generating random elements of a Coxeter group with probability determined by the length function. The analysis is based on interpreting Metropolis walks in terms of the multiplication in the IwahoriHecke algebra. 1.
Cluster Analysis of Heterogeneous Rank Data
"... This revision of the ICML 2007 proceedings article corrects an error in Sec. 3. Cluster analysis of ranking data, which occurs in consumer questionnaires, voting forms or other inquiries of preferences, attempts to identify typical groups of rank choices. Empirically measured rankings are often inco ..."
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Cited by 34 (0 self)
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This revision of the ICML 2007 proceedings article corrects an error in Sec. 3. Cluster analysis of ranking data, which occurs in consumer questionnaires, voting forms or other inquiries of preferences, attempts to identify typical groups of rank choices. Empirically measured rankings are often incomplete, i.e. different numbers of filled rank positions cause heterogeneity in the data. We propose a mixture approach for clustering of heterogeneous rank data. Rankings of different lengths can be described and compared by means of a single probabilistic model. A maximum entropy approach avoids hidden assumptions about missing rank positions. Parameter estimators and an efficient EM algorithm for unsupervised inference are derived for the ranking mixture model. Experiments on both synthetic data and realworld data demonstrate significantly improved parameter estimates on heterogeneous data when the incomplete rankings are included in the inference process. 1.
Consensus ranking under the exponential model
 In Conf. on Uncertainty in Artificial Intelligence (UAI
, 2007
"... Assume that we are given a set of N rankings, a.k.a linear orderings on n objects. For instance, the rankings represent the individual preferences of a panel of N judges, each presented with the same set of n candidate objects. The problem of rank aggregation or of finding a consensus ranking, is fo ..."
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Cited by 34 (5 self)
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Assume that we are given a set of N rankings, a.k.a linear orderings on n objects. For instance, the rankings represent the individual preferences of a panel of N judges, each presented with the same set of n candidate objects. The problem of rank aggregation or of finding a consensus ranking, is formulated as finding a single ranking π0 that best agrees with all the N rankings. 1 Kendall’s correlation [Fligner and Verducci, 1986] is a widely used models of agreement [Ailon et al., 2005, Lebanon and Lafferty, 2003, Cohen et al., 1999]. The Kendall distance is defined as dK(π, π0) = ∑ l≺πj 1 [j≺π0 l] (1) In the above, π, π0 represent permutations and i ≺π j (i ≺π0 j) mean that l precedes j (i.e is preferred to j) in permutation π (π0). Hence dK is the total number of pairwise disagreements between π and π0. This distance was further generalized to a family of parametrized distances [Fligner and Verducci, 1986] by dθ(π, π0) = ∑n−1 −1 j=1 θjVj(ππ0 are given functions on the ranking poset. A
Ordinal measures for visual correspondence
 Columbia Univ. Center for
, 1996
"... We present ordinal measures for establishing ima e correspondence. Linear correspondence measures d e correlation and the sum of squared differences are known to be fragile. Ordinal measures, which are based on relative ordering of intensit values an windows, have demonstrable robustness to Apth dis ..."
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Cited by 33 (5 self)
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We present ordinal measures for establishing ima e correspondence. Linear correspondence measures d e correlation and the sum of squared differences are known to be fragile. Ordinal measures, which are based on relative ordering of intensit values an windows, have demonstrable robustness to Apth discontinuities, occlusion and noise. The relative ordering of intensaty values in each window as represented by a rank permutation which is obtained by sortin the corresponding intensity data. By uszng a novel &stance metric between the rank permutations, we arrive at ordinal correlation coefficients. These coefficients are independent of absolute intensity scale, i.e they are normalized measures. Further, since rank permutations are invariant to monotone transformations of the intensity values, the coefficients are unaffected by nonlinear effects like gamma variation between images. We have developed a simple dgomthm for their eficient implementation. Experiments suggest the superiority of ordinal measures over existing techni ues under nonideal conditions. Though we present orjanal measures in the context o stereo, they serue as a a eneral tool for image matc f ing that is applicable to otter vision problems such as motion estimation and image registratton. 1
Fourier Theoretic Probabilistic Inference over Permutations
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2009
"... Permutations are ubiquitous in many realworld problems, such as voting, ranking, and data association. Representing uncertainty over permutations is challenging, since there are n! possibilities, and typical compact and factorized probability distribution representations, such as graphical models, ..."
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Cited by 30 (8 self)
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Permutations are ubiquitous in many realworld problems, such as voting, ranking, and data association. Representing uncertainty over permutations is challenging, since there are n! possibilities, and typical compact and factorized probability distribution representations, such as graphical models, cannot capture the mutual exclusivity constraints associated with permutations. In this paper, we use the “lowfrequency” terms of a Fourier decomposition to represent distributions over permutations compactly. We present Kronecker conditioning, a novel approach for maintaining and updating these distributions directly in the Fourier domain, allowing for polynomial time bandlimited approximations. Low order Fourierbased approximations, however, may lead to functions that do not correspond to valid distributions. To address this problem, we present a quadratic program defined directly in the Fourier domain for projecting the approximation onto a relaxation of the polytope of legal marginal distributions. We demonstrate the effectiveness of our approach on a real camerabased multiperson tracking scenario.
Nonparametric modeling of partially ranked data
 Journal of Machine Learning Research
"... Statistical models on full and partial rankings of n items are often of limited practical use for large n due to computational consideration. We explore the use of nonparametric models for partially ranked data and derive computationally efficient procedures for their use for large n. The derivatio ..."
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Cited by 29 (3 self)
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Statistical models on full and partial rankings of n items are often of limited practical use for large n due to computational consideration. We explore the use of nonparametric models for partially ranked data and derive computationally efficient procedures for their use for large n. The derivations are largely possible through combinatorial and algebraic manipulations based on the lattice of partial rankings. A biasvariance analysis and an experimental study demonstrate the applicability of the proposed method.
Unsupervised Rank Aggregation with DistanceBased Models
"... The need to meaningfully combine sets of rankings often comes up when one deals with ranked data. Although a number of heuristic and supervised learning approaches to rank aggregation exist, they require domain knowledge or supervised ranked data, both of which are expensive to acquire. In order to ..."
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Cited by 27 (7 self)
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The need to meaningfully combine sets of rankings often comes up when one deals with ranked data. Although a number of heuristic and supervised learning approaches to rank aggregation exist, they require domain knowledge or supervised ranked data, both of which are expensive to acquire. In order to address these limitations, we propose a mathematical and algorithmic framework for learning to aggregate (partial) rankings without supervision. We instantiate the framework for the cases of combining permutations and combining topk lists, and propose a novel metric for the latter. Experiments in both scenarios demonstrate the effectiveness of the proposed formalism. 1.
Comparing partial rankings
 SIAM Journal on Discrete Mathematics
, 2004
"... Abstract. We provide a comprehensive picture of how to compare partial rankings, that is, rankings that allow ties. We propose several metrics to compare partial rankings and prove that they are within constant multiples of each other. Key words. partial ranking, bucket order, permutation, metric AM ..."
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Cited by 26 (2 self)
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Abstract. We provide a comprehensive picture of how to compare partial rankings, that is, rankings that allow ties. We propose several metrics to compare partial rankings and prove that they are within constant multiples of each other. Key words. partial ranking, bucket order, permutation, metric AMS subject classifications. 06A06, 68R99 DOI. 10.1137/05063088X
Conditional Models on the Ranking Poset
 NIPS
, 2002
"... A distancebased conditional model on the ranking poset is presented for use in classification and ranking. The model is an extension of the Mallows model, and generalizes the classifier combination methods used by several ensemble learning algorithms, including error correcting output codes, d ..."
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Cited by 26 (1 self)
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A distancebased conditional model on the ranking poset is presented for use in classification and ranking. The model is an extension of the Mallows model, and generalizes the classifier combination methods used by several ensemble learning algorithms, including error correcting output codes, discrete AdaBoost, logistic regression and cranking. The algebraic structure of the ranking poset leads to a simple Bayesian interpretation of the conditional model and its special cases. In addition to a unifying view, the framework suggests a probabilistic interpretation for error correcting output codes and an extension beyond the binary coding scheme.
Exchangeable pairs and Poisson approximation
 Probab. Surv
, 2005
"... This is a survery paper on Poisson approximation using Stein’s method of exchangeable pairs. We illustrate using Poissonbinomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collector’s problem, and the birthday problem. While ..."
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Cited by 25 (7 self)
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This is a survery paper on Poisson approximation using Stein’s method of exchangeable pairs. We illustrate using Poissonbinomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collector’s problem, and the birthday problem. While many details are new, the results are closely related to a body of work developed by Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry Goldstein, and their collaborators. Some comparison with these other approaches is offered. 1