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We solve tensor balancing, rescaling an Nth order nonnegative tensor by multiplying N tensors of order N - 1 so that every fiber sums to one.
On the foundations of combinatorial theory I: Theory of Möbius functions
G.-C. Rota · 1964
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A relationship between arbitrary positive matrices and doubly stochastic matrices
R. Sinkhorn · 1964
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Concerning nonnegative matrices and doubly stochastic matrices
R. Sinkhorn and P. Knopp · 1967
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Scaling of matrices to achieve specified row and column sums
A. W. Marshall and I. Olkin · 1968
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Forecasts of input-output matrices using the R.A.S. method
A. Parikh · 1979
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An iterative row-action method for interval convex programming
Y. Censor and A. Lent · 1981
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Bregman’s balancing method
B. Lamond and N. F. Stewart · 1981
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Methods for scaling to doubly stochastic form
B. N. Parlett and T. L. Landis · 1982
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The rate of convergence of sinkhorn balancing
G. W. Soules · 1991
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Encyclopedic Dictionary of Mathematics
K. Ito, editor · 1993
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Information geometry on hierarchy of probability distributions
S. Amari · 2001
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Information-geometric measure for neural spikes
H. Nakahara and S. Amari · 2002
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Polyhedral Cones of Magic Cubes and Squares , volume 25 of Algorithms and Combinatorics , pages 25–41
M. Ahmed, J. De Loera, and R. Hemmecke · 2003
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Continuous Lattices and Domains
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott · 2003
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Gene interaction in DNA microarray data is decomposed by information geometric measure
H. Nakahara, S. Nishimura, M. Inoue, G. Hori, and S. Amari · 2003
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Biproportional techniques in input-output analysis: Table updating and structural analysis
M. Lahr and L. de Mesnard · 2004
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Scaling by binormalization
O. E. Livne and G. H. Golub · 2004
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A comparison of descriptive models of a single spike train by information-geometric measure
H. Nakahara, S. Amari, and B. J. Richmond · 2006
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Fair majority voting (or how to eliminate gerrymandering)
M. Balinski · 2008
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A fast algorithm for matrix balancing
P. A. Knight and D. Ruiz · 2013
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Information geometry of positive measures and positive-definite matrices: Decomposable dually flat structure
S. Amari · 2014
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Birkhoff–von Neumann theorem for multistochastic tensors
L.-B. Cui, W. Li, and M. K. Ng · 2014
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A 3D map of the human genome at kilobase resolution reveals principles of chromatin looping
S. S. P. Rao, M. H. Huntley, N. C. Durand, E. K. Stamenova, I. D. Bochkov, J. T. Robinson, A. L. Sanborn, I. Machol, A. D. Omer, E. S. Lander, and E. L. Aiden · 2014
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Learning with a Wasserstein loss
C. Frogner, C. Zhang, H. Mobahi, M. Araya, and T. A. Poggio · 2015
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Convolutional Wasserstein distances: Efficient optimal transportation on geometric domains
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The Sinkhorn–Knopp algorithm: Convergence and applications
P. A. Knight · 2008
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Information geometry and its applications: Convex function and dually flat manifold
S. Amari · 2009
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Input-Output Analysis: Foundations and Extensions
R. E. Miller and P. D. Blair · 2009
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Sinkhorn solves sudoku
T. K. Moon, J. H. Gunther, and J. J. Kupin · 2009
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Sparse low-order interaction network underlies a highly correlated and learnable neural population code
E. Ganmor, R. Segev, and E. Schneidman · 2011
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Categorical data analysis
A. Agresti · 2012
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J. Solomon, F. de Goes, G. Peyré, M. Cuturi, A. Butscher, A. Nguyen, T. Du, and L. Guibas · 2015
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Network models and biproportional rounding for fair seat allocations in the UK elections
K. Akartunalı and P. A. Knight · 2016
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Information Geometry and Its Applications
S. Amari · 2016
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Polytopes of stochastic tensors
H. Chang, V. E. Paksoy, and F. Zhang · 2016
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A review of matrix scaling and sinkhorn’s normal form for matrices and positive maps
M. Idel · 2016
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Information decomposition on structured space
M. Sugiyama, H. Nakahara, and K. Tsuda · 2016
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A computational strategy to adjust for copy number in tumor Hi-C data
H.-J. Wu and F. Michor · 2016
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