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.