Representing and exploiting multivariate signals requires capturing relations between variables, which we can represent by graphs. Graph dictionaries allow to describe complex relational information as a sparse sum of simpler structures, but no prior model exists to infer such underlying structure elements from data. We define a novel Graph-Dictionary signal model, where a finite set of graphs characterizes relationships in data distribution as filters on the weighted sum of their Laplacians. We propose a framework to infer the graph dictionary representation from observed node signals, which allows to include a priori knowledge about signal properties, and about underlying graphs and their coefficients. We introduce a bilinear generalization of the primal-dual splitting algorithm to solve the learning problem. We show the capability of our method to reconstruct graphs from signals in multiple synthetic settings, where our model outperforms popular baselines. Then, we exploit graph-dictionary representations in an illustrative motor imagery decoding task on brain activity data, where we classify imagined motion better than standard methods relying on many more features. Our graph-dictionary model bridges a gap between sparse representations of multivariate data and a structured decomposition of sample-varying relationships into a sparse combination of elementary graph atoms.

翻译：表示和利用多元信号需要捕捉变量间的关系，这些关系可以通过图来表示。图字典允许将复杂的关系信息描述为更简单结构的稀疏和，但目前尚无先验模型能从数据中推断此类底层结构元素。我们定义了一种新颖的图字典信号模型，其中有限图集通过对其拉普拉斯矩阵加权和的滤波来表征数据分布中的关系。我们提出了一个从观测节点信号推断图字典表示的框架，该框架允许纳入关于信号特性、底层图及其系数的先验知识。我们引入了原始-对偶分裂算法的双线性推广来解决该学习问题。我们在多种合成设置中展示了该方法从信号重建图的能力，其中我们的模型优于常用基线方法。随后，我们在脑活动数据的运动想象解码任务中利用图字典表示进行说明性实验，该任务中我们使用比标准方法少得多的特征实现了更优的想象运动分类性能。我们的图字典模型弥合了多元数据稀疏表示与样本变化关系结构化分解为基本图原子稀疏组合之间的空白。