The Dual Graph Shift Operator: Identifying the Support of the Frequency Domain

Abstract

Contemporary data is often supported by an irregular structure, which can be conveniently captured by a graph. Accounting for this graph support is crucial to analyze the data, leading to an area known as graph signal processing (GSP). The two most important tools in GSP are the graph shift operator (GSO), which is a sparse matrix accounting for the topology of the graph, and the graph Fourier transform (GFT), which maps graph signals into a frequency domain spanned by a number of graph-related Fourier-like basis vectors. This alternative representation of a graph signal is denominated the graph frequency signal. Several attempts have been undertaken in order to interpret the support of this graph frequency signal, but they all resulted in a one-dimensional interpretation. However, if the support of the original signal is captured by a graph, why would the graph frequency signal have a simple one-dimensional support? Departing from existing work, we propose an irregular support for the graph frequency signal, which we coin dual graph. A dual GSO leads to a better interpretation of the graph frequency signal and its domain, helps to understand how the different graph frequencies are related and clustered, enables the development of better graph filters and filter banks, and facilitates the generalization of classical SP results to the graph domain.

Topics in Graph Signal Processing: Convolution and Modulation

“…We start by considering the effect of shifting a signal in the frequency domain from which we propose a spectral shift M acting on the graph spectral domain [21]. In [22], the authors define a different spectral shift M , see below, that they require to satisfy a number of properties; for example, permutation invariance that is unfortunately seldom verified.…”

“…Remark 6. Reference [22] defines the spectral shift with Λ rather than Λ * in (8). This is undesirable.…”

“…For example, for the time shift (1) (directed cycle graph) with diagonalization (2) and the DFT given in (3), by result (4) our definition 3 leads to the equality A = M as desired. In contrast, with the definition in [22], the time and spectral shifts M = A T = A, reversing the direction of the cycle graph.…”

“…and let x be the input graph signal to P(A). In the graph spectral domain, filtering is modulation (pointwise multiplication) of x by the graph frequency response of the filter y [7,9] (vertex filtering) P(A) • x F − → y x (spectral modulation) (22) (spectral response) y P (Λ) • 1 and…”

“…To study convolution and sampling, we consider a spectral shift M [21]. Reference [22] has introduced previously a different graph shift, see section III for connections between the two. ASP [18,19] considers impulse signals in the algebraic context, but not the alternatives we discuss here.…”