In this paper, we consider the harmonic extension problem, which is widely used in many applications of machine learning. We formulate the harmonic extension as solving a Laplace-Beltrami equation with Dirichlet boundary condition. We use the point integral method (PIM) to solve the Laplace-Beltrami equation. The basic idea of the PIM method is to approximate the Laplace equation using an integral equation, which is easy to be discretized from points. Based on the integral equation, we found that traditional graph Laplacian method (GLM) may fail to approximate the harmonic functions in the classical sense. For the Laplace-Beltrami equation with Dirichlet boundary, we can prove the convergence of the point integral method. The point integral method is also very easy to implement, which only requires a minor modification of the graph Laplacian. One important application of the harmonic extension in machine learning is semi-supervised learning. We run a popular semi-supervised learning algorithm by Zhu et al. over a couple of well-known datasets and compare the performance of the aforementioned approaches. Our experiments show the PIM performs the best. We also apply PIM to an image recovery problem and show it outperforms GLM. Finally, on a model problem of Laplace-Beltrami equation with Dirichlet boundary, we prove the convergence of the point integral method.

In this paper, we show that every $(2^{n-1}+1)$-vertex induced subgraph of the $n$-dimensional cube graph has maximum degree at least $\sqrt{n}$. This result is best possible, and improves a logarithmic lower bound shown by Chung, F\"uredi, Graham and Seymour in 1988. As a direct consequence, we prove that the sensitivity and degree of a boolean function are polynomially related, solving an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.

We study the minimization problem of a non-convex sparsity promoting penalty func- tion, the transformed l1 (TL1), and its application in compressed sensing (CS). The TL1 penalty inter- polates l0 and l1 norms through a nonnegative parameter a ∈ (0, +∞), similar to lp with p ∈ (0, 1], and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem, and discuss the exact recovery of l0 norm minimal solution based on the null space property (NSP). We then prove the stable recovery of l0 norm minimal solution if the sensing matrix A satisfies a restricted isometry property (RIP). We formulated a normalized problem to overcome the lack of scaling property of the TL1 penalty function. For a general sensing matrix A, we show that the support set of a local minimizer corresponds to linearly independent columns of A. Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized con- strained and unconstrained problems in CS. The DCATL1 algorithm involves outer and inner loops of iterations, one time matrix inversion, repeated shrinkage operations and matrix-vector multiplications. The inner loop concerns an l1 minimization problem on which we employ the Alternating Direction Method of Multipliers (ADMM). For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value a = 1, and compare DCATL1 with other CS algorithms on two classes of sensing ma- trices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). Among existing algorithms, the iterated reweighted least squares method based on l1/2 norm is the best in sparse recovery for Gaussian matrices, and the DCA algorithm based on l1 minus l2 penalty is the best for over-sampled DCT matrices. We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with l1 minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix A. DCATL1 is also com- petitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two hyperparameters.

We formulate the Riemannian calculus of the probability set embedded with L^2-Wasserstein metric. This is an initial work of transport information geometry. Our investigation starts with the probability simplex (probability manifold) supported on vertices of a finite graph. The main idea is to embed the probability manifold as a submanifold of the positive measure space with a nonlinear metric tensor. Here the nonlinearity comes from the linear weighted Laplacian operator. By this viewpoint, we establish torsion-free Christoffel symbols, Levi-Civita connections, curvature tensors, and volume forms in the probability manifold by Euclidean coordinates. As a consequence, the Jacobi equation, Laplace-Beltrami and Hessian operators on the probability manifold are derived. These geometric computations are also provided in the infinite-dimensional density space (density manifold) supported on a finite-dimensional manifold. In particular, an identity is given connecting the Baker-{É}mery Γ2 operator (carr{é} du champ it{é}r{é}) by connecting Fisher-Rao information metric and optimal transport metric. Several examples are demonstrated.

We propose to study the Hessian metric of given functional in the space of probability space embedded with L^2–Wasserstein (optimal transport) metric. We name it transport Hessian metric, which contains and extends the classical L^2–Wasserstein metric. We formulate several dynamical systems associated with transport Hessian metrics. Several connections between transport Hessian metrics and math physics equations are discovered. E.g., the transport Hessian gradient flow, including Newton’s flow, formulates a mean-field kernel Stein variational gradient flow; The transport Hessian Hamiltonian flow of negative Boltzmann-Shannon entropy forms the Shallow water’s equation; The transport Hessian gradient flow of Fisher information forms the heat equation. Several examples and closed-form solutions of finite-dimensional transport Hessian metrics and dynamics are presented.