Machine learning and Probability manifold
The probability space emebed with L2-Wasserstein metric exhibits Riemannian structures. We call it probability manifold. We apply the geometry structure of probability manifold for application problems, including evolutionary games, mean field games, machine learning and geometry of finte graphs etc.
L1 Optimal transport
In 1781, Monge proposed the L1 optimal transport distance among histograms. We develop fast methods for it using techniques borrowed in L1 compressive sensing. Our method is very simple, easy to parallelize and can be combined with other regularizations. We use it in applications including partial optimal transport, image segmentation, image alignment and others. It is flexible enough to easily deal with histograms and other features of the data.
Schrodinger equation and bridge problems
In 1966, Nelson derived Schrödinger equation by the diffusion process. Nowadays this approach connects with the theory of optimal transport. We consider similar matters on finite graphs. We propose a discrete Schrödinger related equations from Nelson’s idea and optimal transport. The proposed equation is the Hamiltonian flow in probability manifold.
- Schrodinger bridge from digits 4 to 1.
Mean field games
We compute numerical solutions of some infinitely dimensional Hamilton-Jacobi equations (HJ-PDE) in probability space that are coming from the theory of mean field games. Numerics towards such HJ-PDE was previously almost impossible owing to the incredibly high dimension of the PDE after discretization of a function space. We propose to utilize the Hopf formula, which comes from an optimal control approach. The resulting formula is an optimization problem involving only a finite dimensional PDE constraint which can be computed using a standard finite difference scheme. In particular, our method will provide us a possible way to compute proximal maps of Wasserstein metrics. They are of special importance in computing optimization problems involving Wasserstein metrics. Our techniques have applications in optimal transport, mean field games and optimal control in the space of probability densities.
- Evolution of Population density.
Evolutionary game theory
We propose a new dynamic framework for the finite or infinite player (population) discrete strategy games. By utilizing tools from optimal transportation theory, we derive Fokker-Planck equations of games. Furthermore, we introduce an associated Best-Reply Markov process that models players’ myopic, greedy and uncertainty when making decisions. The model gives rise to a method to rank/select equilibria for both potential and non-potential games.
We design a new fast algorithm for a class of optimal control problems with constraints on both state and control variables. Instead of searching global minimizer(s) from all feasible paths, we consider the subset of paths with the structure of optimal paths. By leveraging these paths, we transfer optimal control problems to a set of finite and different dimensional optimization problems with constraints. Moreover, for each of these finite dimensional subproblems, we apply methods from stochastic differential equations in order to find numerically all possible global minimizers of our original optimal control problem.
- Frogger game/Finite drone's game.