Online discriminative graph learning from multi-class smooth signals

Abstract

Graph signal processing (GSP) is a key tool for satisfying the growing demand for information processing over networks. However, the success of GSP in downstream learning and inference tasks such as supervised classification is heavily dependent on the prior identification of the relational structures. Graphs are natural descriptors of the relationships between entities of complex environments. The underlying graph is not readily detectable in many cases and one has to infer the topology from the observed signals which admit certain regularity over the sought representation. Assuming that signals are smooth over the latent class-specific graph is the notion we build upon. Firstly, we address the problem of graph signal classification by proposing a novel framework for discriminative graph learning. To learn discriminative graphs, we invoke the assumption that signals belonging to each class are smooth with respect to the corresponding graph while maintaining non-smoothness with respect to the graphs corresponding to other classes. Discriminative features are extracted via graph Fourier transform (GFT) of the learned representations and used in downstream learning tasks. Secondly, we extend our work to tackle increasingly dynamic environments and real-time topology inference. To this end, we adopt recent advances of GSP and time-varying convex optimization. We develop a proximal gradient (PG) method which can be adapted to situations where the data are acquired on-the-fly. Beyond discrimination, this is the first work that addresses the problem of dynamic graph learning from smooth signals where the sought network alters slowly. The introduced online framework is guaranteed to track the optimal time-varying batch solution under mild technical conditions. The validation of the proposed frameworks is comprehensively investigated using both synthetic and real data. The proposed classification pipeline outperforms the-state-of-the-art methods when applied to the problem of emotion classification based on electroencephalogram (EEG) data. We also perform network-based analysis of epileptic seizures using electrocorticography (ECoG) records. Moreover, by applying our method to financial data, our approach infers the relationships between the stock-price behavior of leading US companies and the recent events including the COVID-19 pandemic.