In this era of data deluge, we are overwhelmed with massive volumes of extremely complex datasets. Data generated today is complex because it lacks a clear geometric structure, comes in great volumes, and it often contains information from multiple domains. In this thesis, we address these issues and propose two theoretical frameworks to handle such multidomain dataset.

To begin with, we extend the recently developed geometric deep learning framework to multidomain graph signals, e.g., time-varying signals, defining a new type of convolutional layer that will allow us to deal with graph signals defined on top of several domains, e.g., electroencephalograms or traffic networks. After discussing its properties and motivating its use, we show how this operation can be efficiently implemented to run on a GPU and demonstrate its generalization abilities on a synthetic dataset.

Next, we consider the problem of designing sparse sampling strategies for multidomain signals, which can be represented using tensors. To keep the framework general, we do not restrict ourselves to multidomain signals defined on irregular domains. Nonetheless, this particularization to multidomain graph signals is also presented. To do so, we leverage the multidomain structure of tensor signals and propose to acquire samples using a Kronecker-structured sensing function, thereby circumventing the curse of dimensionality. For designing such sensing functions, we develop several low-complexity greedy algorithms based on submodular optimization methods that compute near-optimal sampling sets. To validate the developed theory, we present several numerical examples, ranging from multi-antenna communications to graph signal processing.