The convexity package for Hamiltonian actions on conformal symplectic manifolds

Consider a Hamiltonian action of a compact connected Lie group on a conformal symplectic manifold. We prove a convexity theorem for the moment map under the assumption that the action is of Lee type, which establishes an analog of Kirwan's convexity theorem in conformal symplectic geometry.Comment: 31 pages, 1 figure. Appendix on conformal presymplectic manifolds added. Minor mistakes correcte

Resource-Adaptive Newton's Method for Distributed Learning

Distributed stochastic optimization methods based on Newton's method offer significant advantages over first-order methods by leveraging curvature information for improved performance. However, the practical applicability of Newton's method is hindered in large-scale and heterogeneous learning environments due to challenges such as high computation and communication costs associated with the Hessian matrix, sub-model diversity, staleness in training, and data heterogeneity. To address these challenges, this paper introduces a novel and efficient algorithm called RANL, which overcomes the limitations of Newton's method by employing a simple Hessian initialization and adaptive assignments of training regions. The algorithm demonstrates impressive convergence properties, which are rigorously analyzed under standard assumptions in stochastic optimization. The theoretical analysis establishes that RANL achieves a linear convergence rate while effectively adapting to available resources and maintaining high efficiency. Unlike traditional first-order methods, RANL exhibits remarkable independence from the condition number of the problem and eliminates the need for complex parameter tuning. These advantages make RANL a promising approach for distributed stochastic optimization in practical scenarios

And\^o dilations for a pair of commuting contractions: two explicit constructions and functional models

One of the most important results in operator theory is And\^o's \cite{ando} generalization of dilation theory for a single contraction to a pair of commuting contractions acting on a Hilbert space. While there are two explicit constructions (Sch\"affer \cite{sfr} and Douglas \cite{Doug-Dilation}) of the minimal isometric dilation of a single contraction, there was no such explicit construction of an And\^o dilation for a commuting pair $(T_1,T_2)$ of contractions, except in some special cases \cite{A-M-Dist-Var, D-S, D-S-S}. In this paper, we give two new proofs of And\^o's dilation theorem by giving both Sch\"affer-type and Douglas-type explicit constructions of an And\^o dilation with function-theoretic interpretation, for the general case. The results, in particular, give a complete description of all possible factorizations of a given contraction $T$ into the product of two commuting contractions. Unlike the one-variable case, two minimal And\^o dilations need not be unitarily equivalent. However, we show that the compressions of the two And\^o dilations constructed in this paper to the minimal dilation spaces of the contraction $T_1T_2$, are unitarily equivalent. In the special case when the product $T=T_1T_2$ is pure, i.e., if $T^{* n}\to 0$ strongly, an And\^o dilation was constructed recently in \cite{D-S-S}, which, as this paper will show, is a corollary to the Douglas-type construction. We define a notion of characteristic triple for a pair of commuting contractions and a notion of coincidence for such triples. We prove that two pairs of commuting contractions with their products being pure contractions are unitarily equivalent if and only if their characteristic triples coincide. We also characterize triples which qualify as the characteristic triple for some pair $(T_1,T_2)$ of commuting contractions such that $T_1T_2$ is a pure contraction.Comment: 24 page

Contextual Dictionary Lookup for Knowledge Graph Completion

Knowledge graph completion (KGC) aims to solve the incompleteness of knowledge graphs (KGs) by predicting missing links from known triples, numbers of knowledge graph embedding (KGE) models have been proposed to perform KGC by learning embeddings. Nevertheless, most existing embedding models map each relation into a unique vector, overlooking the specific fine-grained semantics of them under different entities. Additionally, the few available fine-grained semantic models rely on clustering algorithms, resulting in limited performance and applicability due to the cumbersome two-stage training process. In this paper, we present a novel method utilizing contextual dictionary lookup, enabling conventional embedding models to learn fine-grained semantics of relations in an end-to-end manner. More specifically, we represent each relation using a dictionary that contains multiple latent semantics. The composition of a given entity and the dictionary's central semantics serves as the context for generating a lookup, thus determining the fine-grained semantics of the relation adaptively. The proposed loss function optimizes both the central and fine-grained semantics simultaneously to ensure their semantic consistency. Besides, we introduce two metrics to assess the validity and accuracy of the dictionary lookup operation. We extend several KGE models with the method, resulting in substantial performance improvements on widely-used benchmark datasets

Privacy Computing Meets Metaverse: Necessity, Taxonomy and Challenges

Metaverse, the core of the next-generation Internet, is a computer-generated holographic digital environment that simultaneously combines spatio-temporal, immersive, real-time, sustainable, interoperable, and data-sensitive characteristics. It cleverly blends the virtual and real worlds, allowing users to create, communicate, and transact in virtual form. With the rapid development of emerging technologies including augmented reality, virtual reality and blockchain, the metaverse system is becoming more and more sophisticated and widely used in various fields such as social, tourism, industry and economy. However, the high level of interaction with the real world also means a huge risk of privacy leakage both for individuals and enterprises, which has hindered the wide deployment of metaverse. Then, it is inevitable to apply privacy computing techniques in the framework of metaverse, which is a current research hotspot. In this paper, we conduct comprehensive research on the necessity, taxonomy and challenges when privacy computing meets metaverse. Specifically, we first introduce the underlying technologies and various applications of metaverse, on which we analyze the challenges of data usage in metaverse, especially data privacy. Next, we review and summarize state-of-the-art solutions based on federated learning, differential privacy, homomorphic encryption, and zero-knowledge proofs for different privacy problems in metaverse. Finally, we show the current security and privacy challenges in the development of metaverse and provide open directions for building a well-established privacy-preserving metaverse system. For easy access and reference, we integrate the related publications and their codes into a GitHub repository: https://github.com/6lyc/Awesome-Privacy-Computing-in-Metaverse.git.Comment: In Ad Hoc Networks (2024