Representing and exploiting multivariate signals require capturing complex relations between variables. We define a novel Graph-Dictionary signal model, where a finite set of graphs characterizes relationships in data distribution through a weighted sum of their Laplacians. We propose a framework to infer the graph dictionary representation from observed data, along with a bilinear generalization of the primal-dual splitting algorithm to solve the learning problem. Our new formulation allows to include a priori knowledge on signal properties, as well as on underlying graphs and their coefficients. We show the capability of our method to reconstruct graphs from signals in multiple synthetic settings, where our model outperforms previous baselines. Then, we exploit graph-dictionary representations in a motor imagery decoding task on brain activity data, where we classify imagined motion better than standard methods relying on many more features.

翻译：多元信号的表示与利用需要捕捉变量间的复杂关系。本文定义了一种新颖的图-字典信号模型，其中有限图集通过其拉普拉斯矩阵的加权和来刻画数据分布中的关联结构。我们提出一个从观测数据推断图字典表示的框架，并采用原始-对偶分裂算法的双线性推广来解决该学习问题。新模型能够纳入关于信号特性、底层图结构及其系数的先验知识。我们在多种合成场景中展示了该方法从信号重建图的能力，其性能优于现有基线方法。进一步地，我们将图字典表示应用于脑活动数据的运动想象解码任务，在分类想象运动时取得了比依赖更多特征的传统方法更优的结果。