On the scaling of multidimensional matrices

Franklin, Joel; Lorenz, Jens

doi:10.1016/0024-3795(89)90490-4

Cited by 148 publications

(125 citation statements)

References 15 publications

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“…The Sinkhorn algorithm is known to converge at a linear rate [Franklin and Lorenz 1989;Knight 2008], and similar guarantees exist for alternating projection methods [Escalante and Raydan 2011]. These bounds give a rough indicator of the number of iterations needed to compute convolutional distances and derived quantities used in §6.…”

Section: Timing and Numericsmentioning

confidence: 85%

Convolutional wasserstein distances

Goes²,

et al. 2015

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Figure 1: Shape interpolation from a cow to a duck to a torus via convolutional Wasserstein barycenters on a 100×100×100 grid, using the method at the beginning of §7. AbstractThis paper introduces a new class of algorithms for optimization problems involving optimal transportation over geometric domains. Our main contribution is to show that optimal transportation can be made tractable over large domains used in graphics, such as images and triangle meshes, improving performance by orders of magnitude compared to previous work. To this end, we approximate optimal transportation distances using entropic regularization. The resulting objective contains a geodesic distance-based kernel that can be approximated with the heat kernel. This approach leads to simple iterative numerical schemes with linear convergence, in which each iteration only requires Gaussian convolution or the solution of a sparse, pre-factored linear system. We demonstrate the versatility and efficiency of our method on tasks including reflectance interpolation, color transfer, and geometry processing.

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Section: Timing and Numericsmentioning

confidence: 85%

Convolutional wasserstein distances

Goes²,

et al. 2015

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“…Matrix scaling and its convergence has received increased attention in the past several decades. Rates of convergence Knight (2008), Kalantari et al (1997), and Soules (1991), algorithms Knight and Ruiz (2007), Ruiz (2001), and multidimensional scaling Franklin and Lorenz (1989) are just a few related developing areas. We make use of the SinkhornKnopp convergence result for the nonnegative matrices in the following theorem.…”

Section: Convergence Of Offense-defense Modelmentioning

confidence: 99%

Offense-Defense Approach to Ranking Team Sports

Govan¹,

Langville²,

Meyer³

2009

Journal of Quantitative Analysis in Sports

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The rank of an object is its relative importance to the other objects in the set. Often a rank is an integer assigned from the set 1,...,n. A ranking model is a method of determining a way in which the ranks are assigned. Usually a ranking model uses information available on the objects to determine their respective ratings. The most recognized application of ranking is the competitive sports. Numerous ranking models have been created over the years to compute the team ratings for various sports. In this paper we propose a flexible, easily coded, fast, iterative approach we call the Offense-Defense Model (ODM), to generating team ratings. The convergence of the ODM is grounded in the theory of matrix balancing. * Our special thanks go to Luke Ingram who has worked on developing ODM in his Masters thesis with Amy Langville.

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“…The convergence rate of this procedure was first considered by Franklin and Lorenz [11]. They showed that each step in Sinkhorn's method is a contraction map in the Hilbert projective metric.…”

Section: The Permanentmentioning

confidence: 99%

A Deterministic Strongly Polynomial Algorithm for Matrix Scaling and Approximate Permanents

Linial¹,

Samorodnitsky

Wigderson

2000

Combinatorica

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We present a deterministic strongly polynomial algorithm that computes the permanent of a nonnegative n×n matrix to within a multiplicative factor of e n . To this end we develop the first strongly polynomial-time algorithm for matrix scaling -an important nonlinear optimization problem with many applications. Our work suggests a simple new (slow) polynomial time decision algorithm for bipartite perfect matching, conceptually different from classical approaches. A DETERMINISTIC STRONGLY POLYNOMIAL ALGORITHM 547Our approach to this problem is completely different from the previously taken routes. It involves a natural reduction technique between problem instances: scaling. Observe the following linearity of permanents: Multiplying a row or column by a constant c, multiplies the permanent by c as well. More generally, we say that a matrix B is a scaling of A (by positive vectors x, y ∈ ( + ) n ) if B = XAY , where X = diag(x) and Y = diag(y) are diagonal matrices with x (resp. y) on their diagonal (these being the factors that multiply the rows and the columns respectively). As observed,per(A). Thus scaling reductions not only allow us to compute per(A) from per(B), but in fact any k-factor approximation of per(B) efficiently yields the same approximation for per(A).The idea, then, is to scale an input matrix A into a matrix B whose permanent we can efficiently approximate. A natural strategy is to seek an efficient algorithm for scaling A to a doubly stochastic B. For suppose we succeed: the permanent of B is clearly at most 1, and per(B) ≥ n!/n n > e −n by the lower bound of [7,8]. Consequently, per(A) is also approximated to within an e n factor, as claimed.Note that such a scaling may not always exist -when per(A)= 0. Moreover, even if scaling exists, the scaling vectors x, y may have irrational coordinates, so we may have to settle for an approximately doubly stochastic matrix. The scaling algorithm must, therefore, be accompanied by approximate versions of the van der Waerden bound, and indeed we prove results of the following type (see also proposition 5.1).Lemma 1.2. Let B be a nonnegative n × n matrix, in which all row sums are 1, and where no column sum exceeds 1+ 1 n 2 , then per(B) ≥ e −(n+1) .So we want to efficiently scale A to an almost doubly stochastic matrix. This scaling problem that so naturally arose from our considerations, turned out to have been studied in other contexts as well. The next subsection briefly describes these as well as scaling algorithms -old and new. Matrix scaling Background.Our discussion will be restricted to square matrices, though everything generalizes to rectangular matrices (and some of it even to multidimensional arrays).Let r, c ∈ ( + ) n be two positive vectors with r i = c j . A matrix B is an (r, c)-matrix if r and c are the vectors of row and columns sums of B

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On the scaling of multidimensional matrices

Cited by 148 publications

References 15 publications

Convolutional wasserstein distances

Convolutional wasserstein distances

Offense-Defense Approach to Ranking Team Sports

A Deterministic Strongly Polynomial Algorithm for Matrix Scaling and Approximate Permanents

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