Multi-way Graph Signal Processing on Tensors: Integrative analysis of irregular geometries

Graph signal processing (GSP) is an important methodology for studying data residing on irregular structures. As acquired data is increasingly taking the form of multi-way tensors, new signal processing tools are needed to maximally utilize the multi-way structure within the data. In this paper, we review modern signal processing frameworks generalizing GSP to multi-way data, starting from graph signals coupled to familiar regular axes such as time in sensor networks, and then extending to general graphs across all tensor modes. This widely applicable paradigm motivates reformulating and improving upon classical problems and approaches to creatively address the challenges in tensor-based data. We synthesize common themes arising from current efforts to combine GSP with tensor analysis and highlight future directions in extending GSP to the multi-way paradigm. Over the past decade, graph signal processing (GSP) [1] has laid the foundation for generalizing classical Fourier theory as defined on a regular grid, such as time, to handle signals on irregular structures, such as networks. GSP, however, is currently limited to single-way analysis: graph signals are processed independently of one another, thus ignoring the geometry between multiple graph signals. In the coming decade, generalizing GSP to handle multi-way data, represented by multidimensional arrays or tensors, with graphs underlying each axis of the data will be essential for modern signal processing. To introduce the concept of way, consider a network of N sensors each measuring a signal sampled at T time points. On the one hand, classic signal processing treats these signals as a collection of N independent 1D timeseries ignoring the relation structure of the graph. T. Both are single-way perspectives that ignore the underlying geometry of the other way (also referred to as mode).