Abstract
Extending computational harmonic analysis tools from the classical setting of regular lattices to the more general setting of graphs and networks is very important, and much research has been done recently. The generalized Haar–Walsh transform (GHWT) developed by Irion and Saito (2014) is a multiscale transform for signals on graphs, which is a generalization of the classical Haar and Walsh–Hadamard transforms. We propose the extended generalized Haar–Walsh transform (eGHWT), which is a generalization of the adapted time–frequency tilings of Thiele and Villemoes (1996). The eGHWT examines not only the efficiency of graph-domain partitions but also that of “sequency-domain” partitions simultaneously. Consequently, the eGHWT and its associated best-basis selection algorithm for graph signals significantly improve the performance of the previous GHWT with the similar computational cost, O(N log N), where N is the number of nodes of an input graph. While the GHWT best-basis algorithm seeks the most suitable orthonormal basis for a given task among more than (1.5)^N possible orthonormal bases in mathbb {R}^N, the eGHWT best-basis algorithm can find a better one by searching through more than 0.618cdot (1.84)^N possible orthonormal bases in mathbb {R}^N. This article describes the details of the eGHWT best-basis algorithm and demonstrates its superiority using several examples including genuine graph signals as well as conventional digital images viewed as graph signals. Furthermore, we also show how the eGHWT can be extended to 2D signals and matrix-form data by viewing them as a tensor product of graphs generated from their columns and rows and demonstrate its effectiveness on applications such as image approximation.
Highlights
In recent years, research on graphs and networks is experiencing rapid growth due to a confluence of several trends in science and technology; the advent of new sensors, measurement technologies, and social network infrastructure have provided huge opportunities to visualize complicated interconnected network structures, record data of interest at various locations in such networks, analyze such data, and make inferences and diagnostics
We have introduced the extended Generalized Haar–Walsh Transform
After briefly reviewing the previous Generalized Haar–Walsh Transform (GHWT), we have described how the generalized Haar–Walsh transform (GHWT) can be improved with the new best-basis algorithm, which is the generalization of the Thiele–Villemoes algorithm [49] for the graph setting
Summary
Research on graphs and networks is experiencing rapid growth due to a confluence of several trends in science and technology; the advent of new sensors, measurement technologies, and social network infrastructure have provided huge opportunities to visualize complicated interconnected network structures, record data of interest at various locations in such networks, analyze such data, and make inferences and diagnostics. We propose and develop the extended generalized Haar–Walsh transform (eGHWT). The eGHWT and its associated best-basis selection algorithm for graph signals significantly improve the performance of the previous GHWT with the similar computational cost, O(N log N ) where N is the number of nodes of an input graph. 5, we demonstrate the superiority of the eGHWT over the GHWT (including the graph Haar and Walsh bases) using real datasets. We mainly consider undirected weighted simple connected graphs. We discuss several matrices associated with undirected simple graphs. The information in both V and E is captured by the edge weight matrix W (G) ∈ RN×N , where Wi j ≥ 0 is the edge weight between nodes i and j.
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