A fast algorithm for matrix balancing

IMA Journal of Numerical Analysis

Volumn 33, Issue 3, 2013, Pages 1029-1047

  Knight, Philip A ^a   Ruiz, Daniel ^b  

^a UNIVERSITY OF STRATHCLYDE   (United Kingdom)

^b UNIVERSITÉ DE TOULOUSE   (France)

Author keywords

conjugate gradient iteration; doubly stochastic matrix; matrix balancing; Sinkhorn Knopp algorithm

Indexed keywords

STOCHASTIC SYSTEMS;

CONJUGATE GRADIENT ITERATION; DIAGONAL SCALING; DOUBLY STOCHASTIC MATRIX; INNER-OUTER ITERATIONS; MATRIX-VECTOR PRODUCTS; NON-NEGATIVE MATRIX; PRECONDITIONED CONJUGATE GRADIENT METHOD; SINKHORN - KNOPP;

CONJUGATE GRADIENT METHOD;

EID: 84880302753     PISSN: 02724979     EISSN: 14643642     Source Type: Journal    
DOI: 10.1093/imanum/drs019     Document Type: Article

References (27)

1. BACHARACH, M. (1970) Biproportional Matrices & Input-Output Change. Cambridge: Cambridge University Press.
2. BALAKRISHNAN, A., HWANG, I. & TOMLIN, C. J. (2004) Polynomial approximation algorithms for belief matrix maintenance in identity management. Proceedings of the 43rd IEEE Conference on Decision and Control, 5, (Editors) City: Publisher, pp. 4874-4879.
3. BALINSKI, M. (2008) Fair majority voting (or how to eliminate gerrymandering). Amer. Math. Monthly, 115, 97-113.
4. BRADLEY, A. M. (2010) Algorithms for the equilibration of matrices and their application to limited-memory quasi-Newton methods. Ph.D. Thesis, Stanford University, USA.
5. DASZYKOWSKI, M., MOSLETH FRGESTAD, E., GROVE, H., MARTENS, H. & WALCZAK, B. (2009) Matching 2D gel electrophoresis images with Matlab 'Image Processing Toolbox'. Chemometr. Intell. Lab. Syst., 96, 188-195.
6. DEMBO, R. S., EISENSTAT, S. C. & STEIHAUG, T. (1982) Inexact Newton methods. SIAM J. Numer. Anal., 19, 400-408.
7. DUFF, I. S., GRIMES, R. G. & LEWIS, J. G. (1992) Users' guide for the Harwell-Boeing sparse matrix collection (release I). Technical Report RAL 92-086. Rutherford Appleton Laboratory.
8. FRANKLIN, J. & LORENZ, J. (1989) On the scaling of multidimensional matrices. Linear Algebra Appl., 114/115, 717-735.
9. FÜRER, M. (2004) Quadratic convergence for scaling of matrices. Proc. ALENEX and ANALC 2004 (L. Arge, G. F. Italiano & R. Sedgewick eds). City: Publisher pp. 216-223.
10. HESTENES, M. R. & STIEFEL, E. (1952) Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand., 49, 409-436.
11. KALANTARI, B. & KHACHIYAN, L. (1996) On the complexity of nonnegative-matrix scaling. Linear Algebra Appl., 240, 87-103.
12. KELLEY, C. T. (2003) Solving Nonlinear Equations with Newton's Method. Philadelphia: SIAM.
13. KHACHIYAN, L. & KALANTARI, B. (1992) Diagonal matrix scaling and linear programming. SIAM J. Opt., 2, 668-672.
14. KNIGHT, P. A. (2008) The Sinkhorn-Knopp algorithm: convergence and applications. SIMAX, 30, 261-275.
15. KOLOTILINA, L. YU. (2003) Nonsingularity/singularity criteria for nonstrictly block diagonally dominant matrices. Linear Algebra Appl., 359, 133-159.
16. LAMOND, B. & STEWART, N. F. (1981) Bregman's balancing method. Transp. Res. B, 15, 239-248.
17. LINIAL, N., SAMORODNITSKY, A. & WIGDERSON, A. (2000) Deterministic strongly polynomial algorithm for matrix scaling and approximate permanents. Combinatorica, 20, 545-568.
18. LIVNE, O. E. & GOLUB, G. H. (2004) Scaling by binormalization. Numer. Algorithms, 35, 97-120.
19. MARSHALL, A. W. & OLKIN, I. (1968) Scaling of matrices to achieve specified row and column sums. Numer. Math., 12, 83-90.
20. O'LEARY, D. P. (1980) A generalized conjugate gradient algorithm for solving a class of quadratic programming problems. Linear Algebra Appl., 34, 371-399.
21. ORTEGA, J. M. & RHEINBOLDT, W. C. (1970) Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press.
22. PARLETT, B. N. & LANDIS, T. L. (1982) Methods for scaling to doubly stochastic form. Linear Algebra Appl., 48, 53-79.
23. RUIZ, D. & UCAR, B. (2011) A symmetry preserving algorithm for matrix scaling. Technical report RR-7552. INRIA.
24. SCHNEIDER, M. H. (1990) Matrix scaling, entropy minimization, and conjugate duality (II): The dual problem. Math. Prog., 48, 103-124.
25. SINKHORN, R. & KNOPP, P. (1967) Concerning nonnegative matrices and doubly stochastic matrices. Pacific J. Math., 21, 343-348.
26. SOULES, G. W. (1991) The rate of convergence of Sinkhorn balancing. Linear Algebra Appl., 150, 3-40.
27. TAUSSKY, O. (1948) Bounds for characteristic roots of matrices. Duke Math. J., 15, 1043-1044.