Gradient systems associated with probability distributions
type: Item
instance of: scholarly article
title: Gradient systems associated with probability distributions
zbMATH Open document ID: 0811.58036
DOI: 10.1007/BF03167211
author: Yoshimasa Nakamura
published in: Japan Journal of Industrial and Applied Mathematics
publication date: 1995-04-18T00:00:00Z
review text: Gradient systems on manifolds of probability distributions are studied. It is shown that the gradient systems can be linearized by using Amari's dual coordinate system of the original coordinates. This implies that the gradient flows on the manifolds converge to equilibrium points exponentially. It is proven that the gradient systems are completely integrable Hamiltonian systems if the manifolds have even dimensions. In addition, it is shown that the system associated with a Gaussian distribution admits a Lax pair representation. The gradient systems associated with various distributions including both continuous and discrete distributions are then presented explicitly. Finally, an application of the gradient systems to a nonlinear programming problem is discussed.
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 571992
zbMATH Keywords: manifolds of probability distributions
zbMATH Keywords: gradient systems
zbMATH Keywords: Lax pair representation
reviewed by: Gheorghe Zet
MaRDI profile type: Publication
cites work: Differential-geometrical methods in statistics
cites work: The Nonlinear Geometry of Linear Programming. II Legendre Transform Coordinates and Central Trajectories
cites work: Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems
cites work: Q5583564
cites work: Q4067388
cites work: A new polynomial-time algorithm for linear programming
cites work: Q5202838
cites work: The Nonlinear Geometry of Linear Programming. III Projective Legendre Transform Coordinates and Hilbert Geometry
cites work: Finitely many mass points on the line under the influence of an exponential potential -- an integrable system
cites work: Q4281141
cites work: Completely integrable gradient systems on the manifolds of Gaussian and multinomial distributions
cites work: A geometric method in nonlinear programming
description: scientific article; zbMATH DE number 571992