This paper introduces mathematical frameworks that address the challenges of multi-view clustering in federated learning environments. The objective is to integrate optimization techniques based on new objective functions employing heat-kernel coefficients to replace conventional distance metrics with quantum-inspired measures. The proposed frameworks utilize advanced tensor decomposition methods, specifically, PARAFAC2 and Tucker decomposition to efficiently represent high-dimensional, multi-view data while preserving inter-view relationships. The research has yielded the development of four novel algorithms, an efficient federated kernel multi-view clustering (E-FKMVC) model, FedHK-PARAFAC2, FedHK-Tucker, and FedHK-MVC-Person with PARAFAC2 Decomposition (Personalized FedHK-PARAFAC2). The primary objective of these algorithms is to enhance the efficacy of clustering processes while ensuring the confidentiality and efficient communication in federated learning environments. Theoretical analyses of convergence guarantees, privacy bounds, and complexity are provided to validate the effectiveness of the proposed methods. In essence, this paper makes a significant academic contribution to the field of federated multi-view clustering through its innovative integration of mathematical modeling and algorithm design. This approach addresses the critical challenges of data heterogeneity and privacy concerns, paving the way for enhanced data management and analytics in various contexts.

翻译：本文提出了应对联邦学习环境中多视图聚类挑战的数学框架。其目标在于整合基于新目标函数的优化技术，该目标函数采用热核系数以量子启发的度量替代传统距离度量。所提出的框架利用先进的张量分解方法，特别是PARAFAC2和Tucker分解，以高效表示高维多视图数据，同时保持视图间关系。本研究开发了四种新颖算法：一种高效的联邦核多视图聚类（E-FKMVC）模型、FedHK-PARAFAC2、FedHK-Tucker以及基于PARAFAC2分解的FedHK-MVC个性化模型（个性化FedHK-PARAFAC2）。这些算法的主要目标是在确保联邦学习环境中数据机密性和通信效率的同时，提升聚类过程的有效性。本文提供了收敛性保证、隐私界限和复杂度的理论分析，以验证所提方法的有效性。本质上，本文通过数学建模与算法设计的创新性融合，为联邦多视图聚类领域做出了重要的学术贡献。该方法应对了数据异质性和隐私关切等关键挑战，为各种场景下增强数据管理与分析能力开辟了道路。