Inspired by the graph Laplacian and the point integral method, we introduce a novel weighted graph Laplacian method to compute a smooth interpolation function on a point cloud in high dimensional space. The numerical results in semi-supervised learning and image inpainting show that the weighted graph Laplacian is a reliable and efficient interpolation method. In addition, it is easy to implement and faster than graph Laplacian.

In this paper we integrate semi-local patches and the weighted graph Laplacian into the framework of the low dimensional manifold model. This approach is much faster than the original LDMM algorithm. The number of iterations is typically reduced from 100 to 10 and the equations in each step are much easier to solve. This new approach is tested in image inpainting and denoising and the results are better than the original LDMM and competitive with state-of-the-art methods.

We present the application of a low dimensional manifold model (LDMM) on hyperspectral image (HSI) reconstruction. An important property of hyperspectral images is that the patch manifold, which is sampled by the three-dimensional blocks in the data cube, is generally of a low dimensional nature. This is a generalization of low-rank models in that hyperspectral images with nonlinear mixing terms can also fit in this framework. The point integral method (PIM) is used to solve a Laplace-Beltrami equation over a point cloud sampling the patch manifold in LDMM. Both numerical simulations and theoretical analysis show that the sample points constraint is correctly enforced by PIM. The framework is demonstrated by experiments on the reconstruction of both linear and nonlinear mixed hyperspectral images with a significant number of missing voxels and several entirely missing spectral bands.

In this paper, we give an almost complete classication of toric surface codes of dimension less than or equal to 7, according to monomially equivalence. This is a natural extension of our previous work [YZ], [LYZZ]. More pairs of monomially equivalent toric codes constructed from non-equivalent lattice polytopes are discovered. A new phenomenon appears, that is, the monomially non-equivalence of two toric codes C P(10)7and CP(19)7 can be discerned on Fq, for all q  8, except q = 29. This sudden break seems to be strange and interesting. Moreover, the parameters, such as the numbers of codewords with dierent weights, depends on q heavily. More meticulous analyses have been made to have the possible distinct families of reducible polynomials.

Real-time crime forecasting is important. However, accurate prediction of when and where the next crime will happen is difficult. No known physical model provides a reasonable approximation to such a complex system. Historical crime data are sparse in both space and time and the signal of interests is weak. In this work, we first present a proper representation of crime data. We then adapt the spatial temporal residual network on the well represented data to predict the distribution of crime in Los Angeles at the scale of hours in neighborhood-sized parcels. These experiments as well as comparisons with several existing approaches to prediction demonstrate the superiority of the proposed model in terms of accuracy. Finally, we present a ternarization technique to address the resource consumption issue for its deployment in real world. This work is an extension of our short conference proceeding paper [Wang et al, Arxiv 1707.03340].