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.

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.

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.

We propose BinaryRelax, a simple two-phase algorithm, for training deep neural networks with quantized weights. The set constraint that characterizes the quantization of weights is not imposed until the late stage of training, and a sequence of pseudo quantized weights is maintained. Specifically, we relax the hard constraint into a continuous regularizer via Moreau envelope, which turns out to be the squared Euclidean distance to the set of quantized weights. The pseudo quantized weights are obtained by linearly interpolating between the float weights and their quantizations. A continuation strategy is adopted to push the weights towards the quantized state by gradually increasing the regularization parameter. In the second phase, exact quantization scheme with a small learning rate is invoked to guarantee fully quantized weights. We test BinaryRelax on the benchmark CIFAR- 10 and CIFAR-100 color image datasets to demonstrate the superiority of the relaxed quantization approach and the improved accuracy over the state-of-the-art training methods. Finally, we prove the convergence of BinaryRelax under an approximate orthogonality condition.