Abstract
With the growing application of undirected graphs for signal/image processing on graphs and distributed machine learning, we demonstrate that the shift-enabled condition is as necessary for undirected graphs as it is for directed graphs. It has recently been shown that, contrary to the widespread belief that a shift-enabled condition (necessary for any shift-invariant filter to be representable by a graph shift matrix) can be ignored because any non-shift-enabled matrix can be converted to a shift-enabled matrix, such a conversion in general may not hold for a directed graph with non-symmetric shift matrix. This paper extends this prior work, focusing on undirected graphs where the shift matrix is generally symmetric. We show that while, in this case, the shift matrix can be converted to satisfy the original shift-enabled condition, the converted matrix is not associated with the original graph, that is, it does not capture anymore the structure of the graph signal. We show via examples, that a non-shift-enabled matrix cannot be converted to a shift-enabled one and still maintain the topological structure of the underlying graph, which is necessary to facilitate localized signal processing.
Highlights
Graph signal processing (GSP) extends classical digital signal processing (DSP) to signals on graphs, and provides a prospective solution to numerous real-world problems that involve signals defined on topologically complicated domains, such as social networks, point clouds, biological networks, environmental and condition monitoring sensor networks [1]
In the GSP literature, a graph is uniquely described by a ‘‘shift matrix’’ or a ‘‘shift operator’’,1 S [3]–[5], which has been extensively used for time/vertex-domain filter design
CONTRIBUTION Extending our previous work that looked at directed graphs [18], in this paper, we focus on undirected graphs, and illustrate with examples that when the symmetric shift matrix of an undirected graph is non-shift-enabled, the conversion suggested in [3] could lead to a very different graph that does not necessarily capture the structure of the original graph signal
Summary
Graph signal processing (GSP) extends classical digital signal processing (DSP) to signals on graphs, and provides a prospective solution to numerous real-world problems that involve signals defined on topologically complicated domains, such as social networks, point clouds, biological networks, environmental and condition monitoring sensor networks [1]. In the converted graph, sensors that are far apart might be strongly connected, that is, each output at a vertex could be a linear combination of inputs at almost all vertices, filtering in such converted graph will be computationally unaffordable for ‘‘big data’’ graphs in practice which further emphasizes the importance of the shift-enabled condition [18]. In this manuscript we focus on the case when the topology of the graph is given (for example, by communication constraints due to sensor placement and communication protocols applied).
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