Gradient systems associated with probability distributions

  • Yoshimasa Nakamura


Gradient systems on manifolds of various probability distributions are presented. It is shown that the gradient systems can be linearized by the Legendre transformation. It follows that the corresponding flows on the manifolds converge to equilibrium points of potential functions exponentially. It is proved that gradient systems on manifolds of even dimensions are completely integrable Hamiltonian systems. Especially, the gradient system for the Gaussian distribution admits a Lax pair representation.

Key words

manifolds of probability distributions gradient systems Hamiltonian systems Lax pair representation linear and nonlinear programming problems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Amari, Differential-Geometrical Methods in Statistics. Lecture Notes in Statist. Vol. 28, Springer-Verlag, Berlin, 1985.zbMATHGoogle Scholar
  2. [2]
    D.A. Bayer and J.C. Lagarias, The nonlinear geometry of linear programming. II. Legendre transform coordinates and central trajectories. Trans. Amer. Math. Soc.,314 (1989), 527–581.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    R.W. Brockett, Dynamical systems that sort lists, diagonalize matrices and solve linear programming problem. Linear Algebra Appl.,146 (1991), 79–91.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    I.I. Dikin, Iterative solution of problems of linear and quadratic programming. Soviet Math. Dokl.,8 (1967), 674–675.zbMATHGoogle Scholar
  5. [5]
    A. Fujiwara, Dynamical systems on statistical models. Recent Developments and Perspectives in Nonlinear Dynamical Systems (eds. Y. Nakamura, K. Takasaki and K. Nagatomo), RIMS Kokyuroku No. 822, Kyoto Univ., Kyoto 1993, 32–42.Google Scholar
  6. [6]
    M.W. Hirsch, and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Pure Appl. Math., Vol. 5, Academic Press, New York, 1974.zbMATHGoogle Scholar
  7. [7]
    N. Karmarkar, A new polynomial time algorithm for linear programming. Combinatorica,4 (1984), 373–395.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    N. Karmarkar, Riemannian geometry underlying interior-point methods for linear programming. Mathematical Developments Arising from Linear Programming (eds. J.C. Lagarias and M.J. Todd), Contemp. Math., Vol. 114, Amer. Math. Soc., Providence, 1990, 51–75.Google Scholar
  9. [9]
    J.C. Lagarias, The nonlinear geometry of linear programming. III. Projective Legendre transform coordinates and Hilbert geometry. Trans. Amer. Math. Soc.,320 (1990), 193–225.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    J. Moser, Finitely many points on the line under the influence of an exponential potential — An integrable system. Dynamical Systems, Theory and Applications (ed. J. Moser), Lecture Notes in Phys., Vol. 38, Springer-Verlag, Berlin, 1975, 467–497.CrossRefGoogle Scholar
  11. [11]
    Y. Nakamura, Lax equations associated with a least squares problem and compact Lie algebras. Recent Developments in Differential Geometry (ed. K. Shiohama), Adv. Stud. Pure Math., Vol. 22, Math. Soc. Japan, 1993, 213–229.Google Scholar
  12. [12]
    Y. Nakamura, Completely integrable gradient systems on the manifolds of Gaussian and multinomial distributions. Japan J. Indust. Appl. Math.,10 (1993), 179–189.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    K. Tanabe, A geometric method in nonlinear programming. J. Optim. Theory Appl.,30 (1980), 181–210.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    K. Tanabe and T. Tsuchiya, New geometry in linear programming (in Japanese). Math. Sci., No. 303, 1988, 32–37.Google Scholar

Copyright information

© JJIAM Publishing Committee 1994

Authors and Affiliations

  • Yoshimasa Nakamura
    • 1
  1. 1.Applied Mathematics Laboratory, Department of ElectronicsDoshisha UniversityKyotoJapan

Personalised recommendations