By introducing the notions of L-spaces and L_r-spaces, a complete generalization of Kalton's closed graph theorem is obtained. It points out the class of L_r-spaces is the maximal class of range spaces for the clo...By introducing the notions of L-spaces and L_r-spaces, a complete generalization of Kalton's closed graph theorem is obtained. It points out the class of L_r-spaces is the maximal class of range spaces for the closed graph theorem when the class of domain spaces is the class of Mackey spaces with weakly * sequentially complete dual.Some examples are constructed showing that the class of L_r-spaces is strictly larger than the class of separable B_r-complete spaces.Some properties of L-spaces and L_r-spaces are discussed and the relations between B-complete (resp. B_r-complete) spaces and L-spaces (resp. L_r-spaces) are given.展开更多
In recent years,research has increasingly transformed data into graph representations,using graph neural networks to extract rich relationships and interaction information.This enhances the model’s ability to underst...In recent years,research has increasingly transformed data into graph representations,using graph neural networks to extract rich relationships and interaction information.This enhances the model’s ability to understand and process complex data structures.Due to the privacy and sensitivity of certain data,especially in government and enterprise fields,these high-quality data are often strictly controlled,limiting centralized model training.These issues lead to weaker generalization of traditional models for unseen data.To address these challenges,this paper proposes a Reinforced Federated Graph Domain Generalization(RFGDG)method,which improves generalization across domain data scenarios while protecting data privacy through multi-party collaboration.We design a mini-batch processing strategy based on graph sampling,combined with GraphSage,to build an efficient local graph-based node classification model.This sampling strategy reduces computational overhead while preserving graph structure,improving local model performance.To address data heterogeneity and feature inconsistency across clients,we propose a federated graph domain generalization strategy based on random Fourier feature transformation and weighted covariance matrix optimization,which unifies feature representations,reduces redundancy,and enhances adaptability to inconsistent data.We also propose a dynamic parameter aggregation strategy for federated graph neural networks using deep reinforcement learning.With the Deep Deterministic Policy Gradient(DDPG)algorithm,we dynamically adjust aggregation weights based on each client’s contribution,improving global model accuracy and convergence speed.This strategy considers graph structure heterogeneity and client contribution differences,ensuring generalization in multi-client environments.Extensive experiments on three public graph datasets and one dataset from the Weibo platform demonstrate that the proposed RFGDG method significantly improves global model accuracy and shows stronger robustness and adaptability in multi-client environments.展开更多
In the design of certain kinds of electronic circuits the following question arises:given a non-negative integer k, what graphs admit of a plane embedding such that every edge is a broken lineformed by horizontal and ...In the design of certain kinds of electronic circuits the following question arises:given a non-negative integer k, what graphs admit of a plane embedding such that every edge is a broken lineformed by horizontal and vertical segments and having at mort k bends? Any such graph is said tobe k--rectilinear. No matter what k is, an obvious necessary condition for k-rectilinearity is that thedegree of each vertex does not exceed four.Our main result is that every planar graph H satisfying this condition is 3--rectilinear:in fact,it is 2--rectilinear with the only exception of the octahedron. We also outline a polynomial-timealgorithm which actually constructs a plane embedding of H with at most 2 bends (3 bends if H isthe octahedron) on each edge. The resulting embedding has the property that the total number ofbends does not exceed 2n, where n is the number of vertices of H.展开更多
文摘By introducing the notions of L-spaces and L_r-spaces, a complete generalization of Kalton's closed graph theorem is obtained. It points out the class of L_r-spaces is the maximal class of range spaces for the closed graph theorem when the class of domain spaces is the class of Mackey spaces with weakly * sequentially complete dual.Some examples are constructed showing that the class of L_r-spaces is strictly larger than the class of separable B_r-complete spaces.Some properties of L-spaces and L_r-spaces are discussed and the relations between B-complete (resp. B_r-complete) spaces and L-spaces (resp. L_r-spaces) are given.
基金supported by the National Natural Science Foundation of China(Grant Nos.U23A20319 and 62172056)Young Elite Scientists Sponsorship Program by CAST(2022QNRC001).
文摘In recent years,research has increasingly transformed data into graph representations,using graph neural networks to extract rich relationships and interaction information.This enhances the model’s ability to understand and process complex data structures.Due to the privacy and sensitivity of certain data,especially in government and enterprise fields,these high-quality data are often strictly controlled,limiting centralized model training.These issues lead to weaker generalization of traditional models for unseen data.To address these challenges,this paper proposes a Reinforced Federated Graph Domain Generalization(RFGDG)method,which improves generalization across domain data scenarios while protecting data privacy through multi-party collaboration.We design a mini-batch processing strategy based on graph sampling,combined with GraphSage,to build an efficient local graph-based node classification model.This sampling strategy reduces computational overhead while preserving graph structure,improving local model performance.To address data heterogeneity and feature inconsistency across clients,we propose a federated graph domain generalization strategy based on random Fourier feature transformation and weighted covariance matrix optimization,which unifies feature representations,reduces redundancy,and enhances adaptability to inconsistent data.We also propose a dynamic parameter aggregation strategy for federated graph neural networks using deep reinforcement learning.With the Deep Deterministic Policy Gradient(DDPG)algorithm,we dynamically adjust aggregation weights based on each client’s contribution,improving global model accuracy and convergence speed.This strategy considers graph structure heterogeneity and client contribution differences,ensuring generalization in multi-client environments.Extensive experiments on three public graph datasets and one dataset from the Weibo platform demonstrate that the proposed RFGDG method significantly improves global model accuracy and shows stronger robustness and adaptability in multi-client environments.
文摘In the design of certain kinds of electronic circuits the following question arises:given a non-negative integer k, what graphs admit of a plane embedding such that every edge is a broken lineformed by horizontal and vertical segments and having at mort k bends? Any such graph is said tobe k--rectilinear. No matter what k is, an obvious necessary condition for k-rectilinearity is that thedegree of each vertex does not exceed four.Our main result is that every planar graph H satisfying this condition is 3--rectilinear:in fact,it is 2--rectilinear with the only exception of the octahedron. We also outline a polynomial-timealgorithm which actually constructs a plane embedding of H with at most 2 bends (3 bends if H isthe octahedron) on each edge. The resulting embedding has the property that the total number ofbends does not exceed 2n, where n is the number of vertices of H.
