4 papers · 1 filter
A Unified Variational Framework for Optimal Transport with Lagrangian Costs
Hailiang Liu
We investigate optimal transport distances induced by general Lagrangian action functionals. Extending the classical Monge - Kantorovich and Benamou - Brenier theories, we derive a…
A Generalized Energy-Based Adaptive Gradient Method for Optimization
Lin Feng, Hailiang Liu
Adaptive Gradient Descent with Energy (AEGD) is a variant of Gradient Descent (GD) designed to address step size sensitivity through an energy-based formulation. AEGD is notable fo…
Variational structure of Fokker-Planck equations with variable mobility
Hailiang Liu, Athanasios E. Tzavaras
We study Fokker--Planck equations with symmetric, positive definite mobility matrices capturing diffusion in heterogeneous environments. A weighted Wasserstein metric is introduced…
Adaptive Preconditioned Gradient Descent with Energy
Hailiang Liu, Levon Nurbekyan, Xuping Tian +1
We propose an adaptive step size with an energy approach for a suitable class of preconditioned gradient descent methods. We focus on settings where the preconditioning is applied…