We consider the problem of inferring graph topology from smooth graph signals in a novel but practical scenario where data are located in distributed clients and prohibited from leaving local clients due to factors such as privacy concerns. The main difficulty in this task is how to exploit the potentially heterogeneous data of all clients under data silos. To this end, we first propose an auto-weighted multiple graph learning model to jointly learn a personalized graph for each local client and a single consensus graph for all clients. The personalized graphs match local data distributions, thereby mitigating data heterogeneity, while the consensus graph captures the global information. Moreover, the model can automatically assign appropriate contribution weights to local graphs based on their similarity to the consensus graph. We next devise a tailored algorithm to solve the induced problem, where all raw data are processed locally without leaving clients. Theoretically, we establish a provable estimation error bound and convergence analysis for the proposed model and algorithm. Finally, extensive experiments on synthetic and real data are carried out, and the results illustrate that our approach can learn graphs effectively in the target scenario.

翻译：我们考虑在一种新颖且实际场景中从平滑图信号推断图拓扑的问题，该场景中数据分布在多个客户端，且由于隐私等因素禁止数据离开本地客户端。此任务的主要难点在于如何在数据孤岛下利用所有客户端可能异构的数据。为此，我们首先提出一种自动加权多图学习模型，该模型联合为每个本地客户端学习个性化图，并为所有客户端学习一个共识图。个性化图匹配本地数据分布，从而减轻数据异质性，而共识图则捕捉全局信息。此外，该模型能根据本地图与共识图的相似度自动为其分配适当的贡献权重。接着，我们设计了一种定制算法来求解所引出的问题，所有原始数据均在本地客户端处理而不离开客户端。理论上，我们为所提出的模型和算法建立了可证明的估计误差界和收敛性分析。最后，我们在合成数据和真实数据上进行了大量实验，结果表明我们的方法能够在目标场景中有效学习图。