Spread and Sparse: Learning Interpretable Transforms for Bandlimited Signals on Directed Graphs

We address the problem of learning a sparsifying graph Fourier transform (GFT) for compressible signals on directed graphs (digraphs). Blending the merits of Fourier and dictionary learning representations, the goal is to obtain an orthonormal basis that captures spread modes of signal variation with respect to the underlying network topology, and yields parsimonious representations of bandlimited signals. Accordingly, we learn a data-adapted dictionary by minimizing a spectral dispersion criterion over the achievable frequency range, along with a sparsity-promoting regularization term on the GFT coefficients of training signals. An iterative algorithm is developed which alternates between minimizing a smooth objective over the Stiefel manifold, and soft-thresholding the graph-spectral domain representations of the signals in the training set. A frequency analysis of temperature measurements recorded across the contiguous United States illustrates the merits of the novel GFT design.

I. Introduction

Network data supported on the vertices of a graph are becoming ubiquitous across disciplines spanning the bio-behavioral sciences and engineering. Examples range from measurements of neural activities at different regions of the brain [1], to vehicle traces over transportation networks [2]. Such data, in a snapshot, can be thought of as graph signals represented by vectors indexed by the N nodes of g. In this context, the goal of graph signal processing (GSP) is to broaden the scope of traditional signal and information processing by developing algorithms that fruitfully exploit the complex relational structure of said signals. Accordingly, generalizations of signal processing tasks have been explored in the literature; see [3] for a recent tutorial treatment.