Sampling and Learning Mallows and Generalized Mallows Models Under the Cayley Distance

Irurozki, Ekhiñe; Calvo, B.; Lozano, José A.

doi:10.1007/s11009-016-9506-7

Cited by 19 publications

(15 citation statements)

References 34 publications

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“…In this sense, performing a proper parameter tuning, and exploring new strategies to adjust the smoothing parameter θ, in a similar manner to the temperature parameter in simulated annealing, could provide further improvements. Finally, the work in this paper could be extended to other problems, such as the linear ordering problem [12], and also to other distances, such as Ulam [21] or Hamming [20]. …”

Section: Discussionmentioning

confidence: 99%

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Kernels of Mallows Models for Solving Permutation-based Problems

Ceberio

Mendiburu

Lozano

2015

Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation

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Recently, distance-based exponential probability models, such as Mallows and Generalized Mallows, have demonstrated their validity in the context of estimation of distribution algorithms (EDAs) for solving permutation problems. However, despite their successful performance, these models are unimodal, and therefore, they are not flexible enough to accurately model populations with solutions that are very sparse with regard to the distance metric considered under the model.In this paper, we propose using kernels of Mallows models under the Kendall's-τ and Cayley distances within EDAs. In order to demonstrate the validity of this new algorithm, Mallows Kernel EDA, we compare its performance with the classical Mallows and Generalized Mallows EDAs, on a benchmark of 90 instances of two different types of permutation problems: the quadratic assignment problem and the permutation flowshop scheduling problem. Experimental results reveal that, in most cases, Mallows Kernel EDA outperforms the Mallows and Generalized Mallows EDAs under the same distance. Moreover, the new algorithm under the Cayley distance obtains the best results for the two problems in terms of average fitness and computational time.

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Section: Discussionmentioning

confidence: 99%

“…In addition to the Kendall's-τ distance, recently, another metric on permutations, the Cayley distance, has been introduced in this context [19]. The Cayley distance, Dc(σ, π), counts the minimum number of swaps (not necessary adjacent) that have to be performed to transform σ into π.…”

Section: Kernels Of Mallows Modelsmentioning

confidence: 99%

Kernels of Mallows Models for Solving Permutation-based Problems

Ceberio

Mendiburu

Lozano

2015

Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation

Self Cite

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“…The Kendall distance, which measures the number of adjacent transpositions required to convert R into ρ or, equivalently, the number of discordant pairs in R and ρ, is by far the most popular one in the literature on the MM, mainly for computational reasons. The partition function Z n (α) exists in closed form only for some choices of right-invariant distances, in particular for the Kendall distance (Lu & Boutilier, 2014;Meilǎ & Chen, 2010), the Hamming distance (Irurozki et al, 2014) and the Cayley distance (Irurozki et al, 2018).…”

Section: Methods For Rank Aggregation Inference On the Consensusmentioning

confidence: 99%

Model-Based Learning from Preference Data

Liu

Crispino

Scheel

et al. 2019

Annu. Rev. Stat. Appl.

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Preference data occur when assessors express comparative opinions about a set of items, by rating, ranking, pair comparing, liking, or clicking. The purpose of preference learning is to ( a) infer on the shared consensus preference of a group of users, sometimes called rank aggregation, or ( b) estimate for each user her individual ranking of the items, when the user indicates only incomplete preferences; the latter is an important part of recommender systems. We provide an overview of probabilistic approaches to preference learning, including the Mallows, Plackett–Luce, and Bradley–Terry models and collaborative filtering, and some of their variations. We illustrate, compare, and discuss the use of these methods by means of an experiment in which assessors rank potatoes, and with a simulation. The purpose of this article is not to recommend the use of one best method but to present a palette of different possibilities for different questions and different types of data.

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“…, where the composition operation of the permutation group S n is denoted by • and X j (σ•(σ ) −1 ) = 0 if j is the largest item in its cycle and is equal to 1 otherwise [18]. It is also equal to the minimum number of pairwise transpositions taking σ to σ .…”

Section: Metrics For Permutations and Propertiesmentioning

confidence: 99%

Antithetic and Monte Carlo kernel estimators for partial rankings

et al. 2019

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In the modern age, rankings data is ubiquitous and it is useful for a variety of applications such as recommender systems, multi-object tracking and preference learning. However, most rankings data encountered in the real world is incomplete, which prevents the direct application of existing modelling tools for complete rankings. Our contribution is a novel way to extend kernel methods for complete rankings to partial rankings, via consistent Monte Carlo estimators for Gram matrices: matrices of kernel values between pairs of observations. We also present a novel variance reduction scheme based on an antithetic variate construction between permutations to obtain an improved estimator for the Mallows kernel. The corresponding antithetic kernel estimator has lower variance and we demonstrate empirically that it has a better performance in a variety of Machine Learning tasks. Both kernel estimators are based on extending kernel mean embeddings to the embedding of a set of full rankings consistent with an observed partial ranking. They form a computationally tractable alternative to previous approaches for partial rankings data. An overview of the existing kernels and metrics for permutations is also provided.

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Sampling and Learning Mallows and Generalized Mallows Models Under the Cayley Distance

Cited by 19 publications

References 34 publications

Kernels of Mallows Models for Solving Permutation-based Problems

Kernels of Mallows Models for Solving Permutation-based Problems

Model-Based Learning from Preference Data

Antithetic and Monte Carlo kernel estimators for partial rankings

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