This paper presents a novel framework for tensor eigenvalue analysis in the context of multi-modal data fusion, leveraging topological invariants such as Betti numbers. Traditional approaches to tensor eigenvalue analysis often extend matrix theory, whereas this work introduces a topological perspective to enhance the understanding of tensor structures. By establishing new theorems that link eigenvalues to topological features, the proposed framework provides deeper insights into the latent structure of data, improving both interpretability and robustness. Applications in data fusion demonstrate the theoretical and practical significance of this approach, with potential for broad impact in machine learning and data science.

翻译：本文提出了一种新颖的张量特征值分析框架，用于多模态数据融合场景，并利用了贝蒂数等拓扑不变量。传统的张量特征值分析方法通常是对矩阵理论的扩展，而本研究引入了拓扑视角以增强对张量结构的理解。通过建立连接特征值与拓扑特征的新定理，所提出的框架为数据的潜在结构提供了更深入的见解，从而提升了可解释性与鲁棒性。在数据融合中的应用验证了该方法的理论与实际意义，展现了其在机器学习和数据科学领域的广泛影响潜力。