Signal Representations on Graphs: Tools and Applications

Abstract

We present a framework for representing and modeling data on graphs. Based on this framework, we study three typical classes of graph signals: smooth graph signals, piecewiseconstant graph signals, and piecewise-smooth graph signals. For each class, we provide an explicit definition of the graph signals and construct a corresponding graph dictionary with desirable properties. We then study how such graph dictionary works in two standard tasks: approximation and sampling followed with recovery, both from theoretical as well as algorithmic perspectives. Finally, for each class, we present a case study of a real-world problem by using the proposed methodology.

On Computationally Tractable Selection of Experiments in Measurement-Constrained Regression Models

“…Example 3 (Wind speed prediction) In Chen et al (2015) a data set is created to record wind speed across a year at established measurement locations on highways in Minnesota, US. Due to instrumentation constraints, wind speed can only be measured at intersections of high ways, and a small subset of such intersections is selected for wind speed measurement in order to reduce data gathering costs.…”

Towards Interpretable Sparse Graph Representation Learning with Laplacian Pooling

“…The intuition of using the Laplacian maxima for selecting the centroids is that a smooth signal can be very well approximated using a linear interpolation between its local maxima and minima. This is in contrast with most approaches in GSP that use the lower frequencies for signal conservation, but requires the signal to be k-bandlimited [2,3,27]. For a 1D signal, LaPool selects points, usually near the maxima/minima, where the derivative changes the most and is hardest to interpolate linearly (see Appendix C for further details).…”

“…For DiffPool, we performed a hyperparameter search to find the optimal number of clusters (3,5,7,9). Similarly, a search is also performed for the Graph-Unet pooling layer to determine the best number of node to retain (3,5,7,9). The grid values were set in a way that appropriately reflects the size of the molecules in the datasets.…”

Fractional Spectral Graph Wavelets and Their Applications

“…On the other hand, Fourier transform and wavelet transform were extended to graph domain, obtaining graph Fourier transform (GFT) [33][34][35][36][37][38][39][40] and graph wavelet transform (GWT) [41][42][43][44] to handle signal defined on the vertices of weighted graphs. Two basic approaches to signal processing on graphs have been considered: The first is rooted in the spectral graph theory [45] and builds upon the graph Laplacian matrix [33].…”

“…In particular, the graph Fourier transform in this framework expands a graph signal into a basis of eigenvectors of the adjacency matrix, and the corresponding spectrum is then given by the associated eigenvalues. Besides graph-based transforms [33][34][35][36][37][38][39][40][41][42][43][44], recent research works on graph also include, among others, sampling and interpolation on graphs [48,49], graph signal recovery [50][51][52][53], semi-supervised classification on graphs [54,55], graph dictionary learning [56,57], graph convolutional neural networks [58][59][60][61][62][63][64][65]. Please refer to [66] for more references of graph signal processing.…”