离散几何线上讨论班

• 9:30-10:30 July 6, 2020
  
  Li-Yau inequality and Gaussian estimate for the heat kernel on graphs.
  
  刘双（中国人民大学）
  
  Tencent Meeting ID: 954 492 804, Code: 24680
  
  Abstract: Studying the heat semigroup, we prove Li-Yau inequality for bounded and positive solutions of the heat equation on graphs. These are proved under the assumption of the curvature-dimension inequality CDE'(n,0), which can be considered as a notion of curvature for graphs. We further show that non-negatively curved graphs also satisfy the volume doubling property. From this we prove a Gaussian estimate for the heat kernel, along with Poincare and Harnack inequalities.
  


• 8:30-9:30 June 21, 2020
  
  The heat equation on graphs: unbounded Laplacians, stochastic completeness, intrinsic metrics and volume growth.
  
  Radoslaw Wojciechowski CUNY
  
  Zoom Meeting ID: 666 263 17658 Code: 246802
  
  Abstract: We will discuss the existence and uniqueness (aka stochastic completeness) of solutions of the heat equation on infinite graphs. This will require us to introduce the case of unbounded Laplacians. Furthermore, in order to get volume growth criteria for stochastic completeness which are analogous to the manifold setting, we will need to introduce the concept of an intrinsic metric.
  


• 9:30-10:30 June 14, 2020
  
  Finding the global graph partition via convex relaxation: spectral clustering and community detection.
  
  凌舒畅（上海纽约大学）
  
  Tencent Meeting ID 500 106 721, Code 24680
  
  Abstract: Spectral clustering has become one of the most widely used clustering techniques when the structure of the individual clusters is non-convex or highly anisotropic. Yet, despite its immense popularity, there exists fairly little theory about performance guarantees for spectral clustering. This issue is partly due to the fact that spectral clustering typically involves two steps which complicated its theoretical analysis: first, the eigenvectors of the associated graph Laplacian are used to embed the dataset, and second, k-means clustering algorithm is applied to the embedded dataset to get the labels. This paper is devoted to the theoretical foundations of spectral clustering and graph cuts. We consider a convex relaxation of graph cuts, namely ratio cuts and normalized cuts, that makes the usual two-step approach of spectral clustering obsolete and at the same time gives rise to a rigorous theoretical analysis of graph cuts and spectral clustering. We derive deterministic bounds for successful spectral clustering via a spectral proximity condition that naturally depends on the algebraic connectivity of each cluster and the inter-cluster connectivity. Moreover, we demonstrate by means of some popular examples that our bounds can achieve near-optimality. Our findings are also fundamental to the theoretical understanding of kernel k-means. Numerical simulations confirm and complement our analysis.
  


• 9:30-10:30 June 7, 2020
  
  利用图上的热核估计讨论半线性抛物方程解的爆破问题
  
  吴艺婷中国计量大学
  
  Tencent Meeting ID 224 185 377, Code 24680