Research
Entropy dissipation for underdamped Langevin dynamics
It is of particular interest to study the convergence to equilibrium phenomenon and the explicit convergence rate for degenerate diffusion process. This convergence is perfectly reflected through the Kullback-Leibler-divergence (relative entropy), which measures the distance of two probability distributions. It is equivalent to study the Logarithmic Sobolev inequalities for the desired KL-divergence associated with the stochastic process. The study of the sub-Riemannian Ricci curvature tensor plays a crucial role here. We introduce new angles to define the sub-Riemannian Ricci curvature tensor by using the smallest eigenvalues of certain matrices, which enables us to study the convergence rate in more general settings. Our new methods can be applied to study the LSI for weakly self-consistent Vlasov-Fokker-Planck equations, oscillator chain models with nearest-neighbor couplings, displacement group, etc. (Left: smallest eigen value for variable diffusion Langevin dynamics with specific choice of U and r)
Non-convex learning for big data.
Replica exchange-Monte Carlo is an important technique for accelerating the convergence of the conventional Markov Chain Monte Carlo (MCMC) algorithms. The red chain with high temperature is for exploration of the entire domain, while the blue chain with lower temperature is for exploitation in local region. By swapping the two chains, we aim to find the global minimizer or conduct multi-modal distribution sampling. We apply the algorithm in image classification and uncertainty quantification. My other algorithms involve multiple chain Langevin dynamics and Langevin dynamics on manifolds.
Numerical solutions for Path Dependent PDE and option pricing.
Cubature measrue is constructed to approximate the Wiener measure. The linear path (left) are used to construct the Cubature measure of the Volterra SDE (above) (X,V) by using the Volterra signature. We can numerically compute E[X_T] by using the deterministic paths (left), which approximates the solution of a path-dependent PDE(PPDE).
Deep Signature Forward Backward SDE Algorithm
The truncated signature of a p-variation path at order [p] essentially determines the information of the whole path. The big advantage of this signature method is that this method is not sensitive to the time discretization and could work for large time horizon or high frequency data. We combine the signature method with Deep Neural Network to study state and path dependent FBSDEs. (I.E. parobolic PDE or path-dependent PDE.) We apply our algorithm in European and American option pricing problem.
Horizontal Brownian Motion Paths on Heisenberg group
Left is the horizontal Brownian motion paths on Heisenberg group. One interesting question is to study the quasi-invariance of the horizontal Wiener measure (distribution of the horizontal Brownian motion paths) on sub-Riemannian manifold. We show the quasi-invariance of the horizontal Wiener Measure on totally geodesic foliations (e.g., Hopf fibration). My other work extends to stochastic and geometric analysis on sub-Riemannian manifold.
Hopf Fibration from Wiki
Example: totally geodesic foliation.
Below, I list my research by areas.
Mathematical finance and Machine Learning
(With Erhan Bayraktar, Zhaoyu Zhang). Preprint. arXiv: 2211.11691.
Deep signature FBSDE algorithm. (2022).
(With Man Luo and Zhaoyu Zhang). Numerical Algebra, Control and Optimization. (Accepted).
(With Jianfeng Zhang). SIAM Journal on Financial Mathematics. (Revision)
Non-reversible Parallel Tempering for Uncertainty Approximation in Deep Learning.
(Wei Deng*, Qian Zhang*, Qi Feng*, Faming Liang, Guang Lin).
Thirty-Seventh AAAI Conference on Artificial Intelligence.(Accepted) (2023)
(Wei Deng, Qi Feng, Georgios karagiannis, Guang Lin, Faming Liang.).
Ninth International Conference on Learning Representations. (ICLR, 2021).
(Wei Deng, Qi Feng, Liyao Gao, Faming Liang , Guang Lin).
37th International Conference on Machine Learning,Vienna. (ICML, 2020).
(With Zhenyu Cui, Ruimeng Hu, and Bin Zou). Operations Research Letters.
A Non-linear Zakai-Type Stochastic PDE for Conditional McKean-Vlasov Stochastic Systems.
(With Jin Ma).Preprint.
Density asymptotics for rough volatility models. (2022)
(With Fabrice Baudoin, Cheng Ouyang). Preprint.
Curvature aware langevin dynamics for non-convex sampling. (2022)
(With Wuchen Li, Rongjie Lai). Preprint.
Entropy dissipation and Sub-Riemannian geometry.
(With Erhan Bayraktar, and Wuchen Li). Preprint. arXiv: 2204.1209.
(With Wuchen Li). arXiv: 2102:00544.
Entropy dissipation via Information Gamma calculus: Non-reversible stochastic differential equations.. (2020) (With Wuchen Li). arXiv:2011.08058.
Sub-Riemannian Ricci curvature via generalized Gamma z calculus. (2020).
(With Wuchen Li). arXiv: 2004.01863.
Entropy dissipation for degenerate stochastic differential equations via sub-Riemannian density manifold
(2019). (With Wuchen Li). arXiv: 1910.07480.
Harnack inequalities on totally geodesic foliations with transverse Ricci flow (2017). arXiv:1712.02275.
Rough paths theory and stochastic PDEs.
(With Fabrice Baudoin and Cheng Ouyang). Transactions of the AMS. 2020.
(With Samy Tindel). Stochastic Analysis and Related Topics. pp 117-138. Progress in Probability, vol 72. Birkhäuser, Cham.
(With Xuejing Zhang). Journal of Stochastic Analysis, 1(2) , 4. (2020)
Stochastic differential geometry.
Integration by parts and Quasi-invariance for the horizontal Wiener measure on a foliated compact manifold. (2019). (With Fabrice Baudoin, Maria Gordina). Journal of Functional Analysis. 2019.
Log-Sobolev inequalities on the horizontal path space of a totally geodesic foliation. (2015)
(With Fabrice Baudoin). arXiv:1503.08180.
Ph.D Dissertation: "Topics in Stochastic Analysis and Riemannian Foliations" (2018). Doctoral Dissertations. 1781.