The problem of graph learning concerns the construction of an explicit topological structure revealing the relationship between nodes representing data entities, which plays an increasingly important role in the success of many graph-based representations and algorithms in the field of machine learning and graph signal processing. In this paper, we propose a novel graph learning framework that incorporates prior information along node and observation side, and in particular the covariates that help to explain the dependency structures in graph signals. To this end, we consider graph signals as functions in the reproducing kernel Hilbert space associated with a Kronecker product kernel, and integrate functional learning with smoothness-promoting graph learning to learn a graph representing the relationship between nodes. The functional learning increases the robustness of graph learning against missing and incomplete information in the graph signals. In addition, we develop a novel graph-based regularisation method which, when combined with the Kronecker product kernel, enables our model to capture both the dependency explained by the graph and the dependency due to graph signals observed under different but related circumstances, e.g. different points in time. The latter means the graph signals are free from the i.i.d. assumptions required by the classical graph learning models. Experiments on both synthetic and real-world data show that our methods outperform the state-of-the-art models in learning a meaningful graph topology from graph signals, in particular with heavy noise, missing values, and multiple dependency.

中文翻译：

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从平滑信号进行基于核的图学习：一个功能性的观点

图学习的问题涉及显式拓扑结构的构建，该结构揭示了表示数据实体的节点之间的关系，这在机器学习和图信号处理领域的许多基于图的表示和算法的成功中起着越来越重要的作用。在本文中，我们提出了一种新颖的图学习框架，该框架结合了沿节点和观察端的先验信息，尤其是有助于解释图信号中依存关系的协变量。为此，我们将图信号视为与Kronecker乘积内核关联的再现内核Hilbert空间中的函数，并将功能学习与平滑度提高的图学习相结合，以学习表示节点之间关系的图。功能学习提高了图学习针对图信号中丢失和不完整信息的鲁棒性。此外，我们开发了一种新颖的基于图的正则化方法，当与Kronecker产品内核结合使用时，我们的模型既可以捕获图解释的依赖性，又可以捕获在不同但相关的环境（例如不同）下观察到的图信号引起的依赖性。时间点。后者意味着图信号没有经典图学习模型所要求的iid假设。对合成数据和实际数据进行的实验表明，在从图形信号中学习有意义的图形拓扑时，尤其是在噪声较大，缺少值和存在多个依存关系的情况下，我们的方法优于最新模型。