A computational method, based on l1-minimization, is proposed for the problem of link flow correction, when the available traffic flow data on many links in a road network are inconsistent with respect to the flow conservation law. Without extra information, the problem is generally ill-posed when a large portion of the link sensors are unhealthy. It is possible, however, to correct the corrupted link flows accurately with the proposed method under a recoverability condition if there are only a few bad sensors which are located at certain links. We analytically identify the links that are robust to miscounts and relate them to the geometric structure of the traffic network by introducing the recoverability concept and an algorithm for computing it. The recoverability condition for corrupted links is simply the associated recoverability being greater than 1. In a more realistic setting, besides the unhealthy link sensors, small measurement noises may be present at the other sensors. Under the same recoverability condition, our method guarantees to give an estimated traffic flow fairly close to the ground-truth data and leads to a bound for the correction error. Both synthetic and real-world examples are provided to demonstrate the effectiveness of the proposed method.

We present a new perspective on graph-based methods for collaborative ranking for recommender systems. Unlike user-based or itembased methods that compute a weighted average of ratings given by the nearest neighbors, or lowrank approximation methods using convex optimization and the nuclear norm, we formulate matrix completion as a series of semi-supervised learning problems, and propagate the known ratings to the missing ones on the user-user or itemitem graph globally. The semi-supervised learning problems are expressed as Laplace-Beltrami equations on a manifold, or namely, harmonic extension, and can be discretized by a point integral method. We show that our approach does not impose a low-rank Euclidean subspace on the data points, but instead minimizes the dimension of the underlying manifold. Our method, named LDM (low dimensional manifold), turns out to be particularly effective in generating rankings of items, showing decent computational efficiency and robust ranking quality compared to state-ofthe- art methods.

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.

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.

In this paper, we propose a novel low dimensional manifold model (LDMM) and apply it to some image processing problems. LDMM is based on the fact that the patch manifolds of many natural images have low dimensional structure. Based on this fact, the dimension of the patch manifold is used as a regularization to recover the image. The key step in LDMM is to solve a Laplace-Beltrami equation over a point cloud which is solved by the point integral method. The point integral method enforces the sample point constraints correctly and gives better results than the standard graph Laplacian. Numerical simulations in image denoising, inpainting and super-resolution problems show that LDMM is a powerful method in image processing.