Further notes on Birkhoff–von Neumann decomposition of doubly stochastic matrices

Dufossé, Fanny; Kaya, Kamer; Panagiotas, Ioannis; Uçar, Bora

doi:10.1016/j.laa.2018.05.017

Cited by 12 publications

(3 citation statements)

References 8 publications

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“…Similarly to Wang and Joachims [34], we make use of the fact that the Birkhoff-von Neumann decomposition is not unique. For most doubly stochastic matrices there is a large number of possible decompositions [13], which makes it possible for us to search for a decomposition that does not have a lot of weight on rankings with unknown exposure distribution. After determining the MRP matrix P (line 1), we decompose it into the sum P = 𝑀 𝑖=1 𝛼 𝑖 𝑃 𝜎 𝑖 .…”

Section: Determining a Stochastic Policy That Avoids Rankings With Un...mentioning

confidence: 99%

Fairness of Exposure in Light of Incomplete Exposure Estimation

Heuss,

Sarvi,

de Rijke

2022

Preprint

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Fairness of exposure is a commonly used notion of fairness for ranking systems. It is based on the idea that all items or item groups should get exposure proportional to the merit of the item or the collective merit of the items in the group. Often, stochastic ranking policies are used to ensure fairness of exposure. Previous work unrealistically assumes that we can reliably estimate the expected exposure for all items in each ranking produced by the stochastic policy. In this work, we discuss how to approach fairness of exposure in cases where the policy contains rankings of which, due to inter-item dependencies, we cannot reliably estimate the exposure distribution. In such cases, we cannot determine whether the policy can be considered fair. Our contributions in this paper are twofold. First, we define a method called FELIX for finding stochastic policies that avoid showing rankings with unknown exposure distribution to the user without having to compromise user utility or item fairness. Second, we extend the study of fairness of exposure to the top-𝑘 setting and also assess FELIX in this setting. We find that FELIX can significantly reduce the number of rankings with unknown exposure distribution without a drop in user utility or fairness compared to existing fair ranking methods, both for full-length and top-𝑘 rankings. This is an important first step in developing fair ranking methods for cases where we have incomplete knowledge about the user's behaviour. CCS CONCEPTS• Information systems → Evaluation of retrieval results; Retrieval models and ranking.

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Section: Determining a Stochastic Policy That Avoids Rankings With Un...mentioning

confidence: 99%

Fairness of Exposure in Light of Incomplete Exposure Estimation

Heuss,

Sarvi,

de Rijke

2022

Preprint

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“…While finding the minimum k is shown to be NP-hard [31], by Marcus-Ree theorem, we know that there exists one constructible decomposition where k < (n − 1) 2 + 1.…”

Section: Definition 5 (Doubly Stochastic (Ds) Matrix)mentioning

confidence: 99%

Probabilistic Permutation Synchronization Using the Riemannian Structure of the Birkhoff Polytope

Birdal

Şimşekli

2019

2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)

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We present an entirely new geometric and probabilistic approach to synchronization of correspondences across multiple sets of objects or images. In particular, we present two algorithms: (1) Birkhoff-Riemannian L-BFGS for optimizing the relaxed version of the combinatorially intractable cycle consistency loss in a principled manner, (2) Birkhoff-Riemannian Langevin Monte Carlo for generating samples on the Birkhoff Polytope and estimating the confidence of the found solutions. To this end, we first introduce the very recently developed Riemannian geometry of the Birkhoff Polytope. Next, we introduce a new probabilistic synchronization model in the form of a Markov Random Field (MRF). Finally, based on the first order retraction operators, we formulate our problem as simulating a stochastic differential equation and devise new integrators. We show on both synthetic and real datasets that we achieve high quality multi-graph matching results with faster convergence and reliable confidence/uncertainty estimates.

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“…The problem of decomposing a demand matrix into matchings, i.e., the decomposition of a matrix into permutation matrices, was considered by [3,8,17,22]. The special cases of zero delay [14] and infinite delay [24] have also been considered.…”

Section: Related Workmentioning

confidence: 99%

Online and Offline Greedy Algorithms for Routing with Switching Costs

Schwartz¹,

Singh²,

Yazdanbod³

2019

Preprint

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Motivated by the use of high speed circuit switches in large scale data centers, we consider the problem of circuit switch scheduling. In this problem we are given demands between pairs of servers and the goal is to schedule at every time step a matching between the servers while maximizing the total satisfied demand over time. The crux of this scheduling problem is that once one shifts from one matching to a different one a fixed delay δ is incurred during which no data can be transmitted.For the offline version of the problem we present a (1 − 1 /e − ) approximation ratio (for any constant > 0). Since the natural linear programming relaxation for the problem has an unbounded integrality gap, we adopt a hybrid approach that combines the combinatorial greedy with randomized rounding of a different suitable linear program. For the online version of the problem we present a (bi-criteria) ((e − 1)/(2e − 1) − )-competitive ratio (for any constant > 0 ) that exceeds time by an additive factor of O( δ / ). We note that no uni-criteria online algorithm is possible. Surprisingly, we obtain the result by reducing the online version to the offline one.

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Further notes on Birkhoff–von Neumann decomposition of doubly stochastic matrices

Cited by 12 publications

References 8 publications

Fairness of Exposure in Light of Incomplete Exposure Estimation

Fairness of Exposure in Light of Incomplete Exposure Estimation

Probabilistic Permutation Synchronization Using the Riemannian Structure of the Birkhoff Polytope

Online and Offline Greedy Algorithms for Routing with Switching Costs

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