Gradient systems associated with probability distributions
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.
scientific article; zbMATH DE number 571992
scientific article