The proof of a conjecture on largest Laplacian and signless Laplacian H-eigenvalues of uniform hypergraphs

Let $\mathcal{A(}G\mathcal{)},\mathcal{L(}G\mathcal{)}$ and $\mathcal{Q(}G\mathcal{)}$ be the adjacency tensor, Laplacian tensor and signless Laplacian tensor of uniform hypergraph $G$, respectively. Denote by $\lambda (\mathcal{T})$ the largest H-eigenvalue of tensor $\mathcal{T}$. Let $H$ be a uniform hypergraph, and $H^{\prime}$ be obtained from $H$ by inserting a new vertex with degree one in each edge. We prove that $\lambda(\mathcal{Q(}H^{\prime}\mathcal{)})\leq\lambda(\mathcal{Q(}H\mathcal{)}).$ Denote by $G^{k}$ the $k$th power hypergraph of an ordinary graph $G$ with maximum degree $\Delta\geq2$. We will prove that $\{\lambda(\mathcal{Q(}$ is a strictly decreasing sequence, which imply Conjectrue 4.1 of Hu, Qi and Shao in \cite{HuQiShao2013}. We also prove that $\lambda(\mathcal{Q(}G^{k}\mathcal{)})$ converges to $\Delta$ when $k$ goes to infinity. The definiton of $k$th power hypergraph $G^{k}$ has been generalized as $G^{k,s}.$ We also prove some eigenvalues properties about $\mathcal{A(}G^{k,s}\mathcal{)},$ which generalize some known results. Some related results about $\mathcal{L(}G\mathcal{)}$ are also mentioned

Towards High-Fidelity 3D Face Reconstruction from In-the-Wild Images Using Graph Convolutional Networks

3D Morphable Model (3DMM) based methods have achieved great success in recovering 3D face shapes from single-view images. However, the facial textures recovered by such methods lack the fidelity as exhibited in the input images. Recent work demonstrates high-quality facial texture recovering with generative networks trained from a large-scale database of high-resolution UV maps of face textures, which is hard to prepare and not publicly available. In this paper, we introduce a method to reconstruct 3D facial shapes with high-fidelity textures from single-view images in-the-wild, without the need to capture a large-scale face texture database. The main idea is to refine the initial texture generated by a 3DMM based method with facial details from the input image. To this end, we propose to use graph convolutional networks to reconstruct the detailed colors for the mesh vertices instead of reconstructing the UV map. Experiments show that our method can generate high-quality results and outperforms state-of-the-art methods in both qualitative and quantitative comparisons.Comment: Accepted to CVPR 2020. The source code is available at https://github.com/FuxiCV/3D-Face-GCN

Diffeomorphism Invariance of Geometric Descriptions of Palatini and Ashtekar Gravity

In this paper, we explicitly prove the presymplectic forms of the Palatini and Ashtekar gravity to be zero along gauge orbits of the Lorentz and diffeomorphism groups, which ensures the diffeomorphism invariance of these theories.Comment: Latex, 6 page

Student Community Detection and Recommendation of Customized Paths to Reinforce Academic Success

Educational Data Mining (EDM) is a research area that analyzes educational data and extracts interesting and unique information to address education issues. EDM implements computational methods to explore data for the purpose of studying questions related to educational achievements. A common task in an educational environment is the grouping of students and the identification of communities that have common features. Then, these communities of students may be studied by a course developer to build a personalized learning system, promote effective group learning, provide adaptive contents, etc. The objective of this thesis is to find an approach to detect student communities and analyze students who do well academically with particular sequences of classes in each community. Then, we compute one or more sequences of courses that a student in a community may pursue to higher their chances of obtaining good academic performance