An affine invariant p-th moment measure is defined for a random vector and used to prove sharp moment-entropy inequalities that are more general and stronger than standard moment-entropy inequalities.

We study generalized Hopf–Cole transformations motivated by the Schrödinger bridge problem, which can be seen as a boundary value Hamiltonian system on the Wasserstein space. We prove that generalized Hopf–Cole transformations are symplectic submersions in the Wasserstein symplectic geometry. Many examples, including a Hopf–Cole transformation for the shallow water equations, are given. Based on this transformation, energy splitting inequalities are provided.

Poisson equation on point cloud with Dirichlet boundary condition plays important role in many problems. In this paper, we use the volume constraint proposed by Du et.al to handle the Dirichlet boundary condition in the point integral method for Poisson equation on point cloud. We prove that the solution given by volume constraint converges to the true solution as the point cloud converges to the underlying smooth manifold.

We study the problem of constructing sparse and fast mean revert- ing portfolios. The problem is motivated by convergence trading and formu- lated as a generalized eigenvalue problem with a cardinality constraint [6]. We use a proxy of mean reversion coefficient, the direct Ornstein-Uhlenbeck (OU) estimator, which can be applied to both stationary and nonstationary data. In addition, we introduce three different methods to enforce the sparsity of the solutions. One method uses the ratio of l1 and l2 norms and the other two use l1 norm. We analyze various formulations of the resulting non-convex opti- mization problems and develop efficient algorithms to solve them for portfolio sizes as large as hundreds. By adopting a simple convergence trading strat- egy, we test the performance of our sparse mean reverting portfolios on both synthetic and historical real market data. In particular, the l1 regularization method, in combination with quadratic program formulation as well as differ-ence of convex functions and least angle regression treatment, gives fast and robust performance on large out-of-sample data set. To appear in Journal of Scientific Computing (JOMP).

The Laplace-Beltrami operator (LBO) is a fundamental object associated to Riemannian manifolds, which encodes all intrinsic geometry of the manifolds and has many desirable prop- erties. Recently, we proposed a novel numerical method, Point Integral method (PIM), to discretize the Laplace-Beltrami operator on point clouds [28]. In this paper, we analyze the convergence of Point Integral method (PIM) for Poisson equation with Neumann boundary condition on submanifolds isometrically embedded in Euclidean spaces.