基于稀疏化变换学习的图拓扑推理
Graph topology inference based on sparsifying transform learning
摘要：
基于图的表示在机器学习中起着关键作用。这些表示的基本步骤是将图结构与数据集相关联。在本文中，我们提出了一种方法，旨在找到一个块稀疏表示的图形信号导致一个模块化图的拉普拉斯矩阵承认发现字典作为其特征向量。稀疏性在这里的作用是诱导一个带限表示，或者等价地，一个图的模块结构。所提出的策略由两个优化步骤组成：i）从数据中学习正交稀疏化变换; ii）从变换中恢复拉普拉斯算子，然后恢复拓扑。第一步是通过迭代算法实现的，其交替的中间解以封闭形式表示。第二步通过凸优化方法从稀疏化变换中恢复拉普拉斯矩阵。数值结果证实了所提出的方法在合成数据和真实大脑数据上的有效性，用于通过对癫痫患者进行的实验来推断大脑功能网络。
英文摘要：
Graph-based representations play a key role in machine learning. The fundamental step in these representations is the association of a graph structure to a dataset. In this paper, we propose a method that aims at finding a block sparse representation of the graph signal leading to a modular graph whose Laplacian matrix admits the found dictionary as its eigenvectors. The role of sparsity here is to induce a band-limited representation or, equivalently, a modular structure of the graph. The proposed strategy is composed of two optimization steps: i) learning an orthonormal sparsifying transform from the data; ii) recovering the Laplacian, and then topology, from the transform. The first step is achieved through an iterative algorithm whose alternating intermediate solutions are expressed in closed form. The second step recovers the Laplacian matrix from the sparsifying transform through a convex optimization method. Numerical results corroborate the effectiveness of the proposed methods over both synthetic data and real brain data, used for inferring the brain functionality network through experiments conducted over patients affected by epilepsy.
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