The Sinkhorn–Knopp Algorithm: Convergence and Applications

Knight, Philip A.

doi:10.1137/060659624

Cited by 249 publications

(195 citation statements)

References 17 publications

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“…Nonnegative and positive regularizable graphs have been already studied in the literature, in particular in connection with the balancing problem of a nonnegative matrix A, namely the question of finding diagonal matrices D 1 and D 2 so that all the rows and columns of P = D 1 AD 2 sum to one [8]. Some motivations for achieving this balance, and hence for studying nonnegative and positive regularizable graphs, include interpreting economic data [9], understanding traffic circulation [10], assigning seats fairly after elections [11], matching protein samples [12] and centrality measures in networks [13,1].…”

Section: Our Contributionmentioning

confidence: 99%

Arbitrarily regularizable graphs

Franceschet¹,

Bozzo²

2017

Internet Mathematics

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A graph is regularizable if it is possible to assign weights to its edges so that all nodes have the same degree. Weights can be positive, nonnegative or arbitrary as soon as the regularization degree is not null. Positive and nonnegative regularizable graphs have been thoroughly investigated in the literature. In this work, we propose and study arbitrarily regularizable graphs. In particular, we investigate necessary and sufficient regularization conditions on the topology of the graph and of the corresponding adjacency matrix. Moreover, we study the computational complexity of the regularization problem and characterize it as a linear programming model.

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Section: Our Contributionmentioning

confidence: 99%

Arbitrarily regularizable graphs

Franceschet¹,

Bozzo²

2017

Internet Mathematics

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“…As alluded to earlier, a DSM is a square matrix with rows and columns summing to one. Sinkhorn [40,41] showed that any non-negative square matrix (with full support [22]) can be converted to a DSM by alternating between rescaling its rows and columns to one. Recently, Adams and Zemel [1] examine the use of DSMs as differentiable relaxations of permutation matrices in gradient based optimization problems.…”

Section: Sinkhorn Normalizationmentioning

confidence: 99%

DeepPermNet: Visual Permutation Learning

Cruz

Fernando

Cherian

et al. 2017

2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)

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We present a principled approach to uncover the structure of visual data by solving a novel deep learning task coined visual permutation learning. The goal of this task is to find the permutation that recovers the structure of data from shuffled versions of it. In the case of natural images, this task boils down to recovering the original image from patches shuffled by an unknown permutation matrix. Unfortunately, permutation matrices are discrete, thereby posing difficulties for gradient-based methods. To this end, we resort to a continuous approximation of these matrices using doubly-stochastic matrices which we generate from standard CNN predictions using Sinkhorn iterations. Unrolling these iterations in a Sinkhorn network layer, we propose DeepPermNet, an end-to-end CNN model for this task.The utility of DeepPermNet is demonstrated on two challenging computer vision problems, namely, (i) relative attributes learning and (ii) self-supervised representation learning. Our results show state-of-the-art performance on the Public Figures and OSR benchmarks for (i) and on the classification and segmentation tasks on the PASCAL VOC dataset for (ii).

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“…Equilibration involves finding diagonal matrices D 1 and D 2 (whose diagonals are positive) such that the ith row sum of X = D 1 AD 2 is t i and the jth column sum of X is s j . The problem has many applications (including interpreting economic data (Bacharach 1970), understanding traffic circulation (Lamond and Stewart 1981), mapping the human genome (Rao et al 2014) and ordering nodes in a graph (Knight 2008)), particularly when A is square and X is doubly stochastic. Existence and uniqueness of solutions is well understood (Brualdi 1968;Pukelsheim 2014) and relates to the nonzero pattern of A.…”

Section: Biproportional Roundingmentioning

confidence: 99%

Network models and biproportional rounding for fair seat allocations in the UK elections

Akartunalı

Knight

2016

Ann Oper Res

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Systems for allocating seats in an election offer a number of socially and mathematically interesting problems. We discuss how to model the allocation process as a network flow problem, and propose a wide choice of objective functions and allocation schemes. Biproportional rounding, which is an instance of the network flow problem, is used in some European countries with multi-seat constituencies. We discuss its application to single seat constituencies and the inevitable consequence that seats are allocated to candidates with little local support. However, we show that variants can be selected, such as regional apportionment, to mitigate this problem. In particular, we introduce a parameter based family of methods, which we call balanced majority voting, that can be tuned to meet the public's demand for local and global "fairness". Using data from the 2010 to 2015 UK general elections, we study a variety of network models and implementations of biproportional rounding, and address conditions of existence and uniqueness.

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The Sinkhorn–Knopp Algorithm: Convergence and Applications

Cited by 249 publications

References 17 publications

Arbitrarily regularizable graphs

Arbitrarily regularizable graphs

DeepPermNet: Visual Permutation Learning

Network models and biproportional rounding for fair seat allocations in the UK elections

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