Uncertainty Principles and Sparse Eigenvectors of Graphs

Abstract

Analysis of signals defined over graphs has been of interest in the recent years. In this regard, many concepts from the classical signal processing theory have been extended to the graph case, including uncertainty principles that study the concentration of a signal on a graph and in its graph Fourier basis (GFB). This paper advances a new way to formulate the uncertainty principle for signals defined over graphs, by using a nonlocal measure based on the notion of sparsity. To be specific, the total number of nonzero elements of a graph signal and its corresponding graph Fourier transform (GFT) is considered. A theoretical lower bound for this total number is derived, and it is shown that a nonzero graph signal and its GFT cannot be arbitrarily sparse simultaneously. When the graph has repeated eigenvalues, the GFB is not unique. Since the derived lower bound depends on the selected GFB, a method that constructs a GFB with the minimal uncertainty bound is provided. In order to find signals that achieve the derived lower bound (i.e., the most compact on the graph and in the GFB), sparse eigenvectors of the graph are investigated. It is shown that a connected graph has a 2-sparse eigenvector (of the graph Laplacian) when there exist two nodes with the same neighbors. In this case, the uncertainty bound is very low, tight, and independent of the global structure of the graph. For several examples of classical and real-world graphs, it is shown that 2-sparse eigenvectors, in fact, exist.