Partition of Unity Methods for Signal Processing on Graphs

Abstract

Partition of unity methods (PUMs) on graphs are simple and highly adaptive auxiliary tools for graph signal processing. Based on a greedy-type metric clustering and augmentation scheme, we show how a partition of unity can be generated in an efficient way on graphs. We investigate how PUMs can be combined with a local graph basis function (GBF) approximation method in order to obtain low-cost global interpolation or classification schemes. From a theoretical point of view, we study necessary prerequisites for the partition of unity such that global error estimates of the PUM follow from corresponding local ones. Finally, properties of the PUM as cost-efficiency and approximation accuracy are investigated numerically. KeywordsPartition of unity method (PUM) • Graph signal processing • Graph basis functions (GBFs) • Kernel-based approximation and interpolation • Spectral graph theory B Wolfgang Erb

Graph Wedgelets: Adaptive Data Compression on Graphs Based on Binary Wedge Partitioning Trees and Geometric Wavelets

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“…As soon as j, or equivalently, q j are determined, a non-adaptive way to choose the subsequent node set q m+1 is by the selection rule q m+1 = arg max v∈V (m) q j d(q j , v), i.e., q m+1 is the vertex in V (m) q j furthest away from q j . This choice and the corresponding split can be interpreted as a two center clustering of V (m) q j in which the first node q j is fixed (see a previous work [3] for more details on greedy J-center clustering). One heuristic reason for this selection is that the resulting binary partitions in the BWP tree might be more balanced with a smaller constant ρ compared to the theoretical upper bound 1 − 1/n in Proposition IV.…”

Krylov subspace methods to accelerate kernel machines on graphs

“…Remark 1. (i) An alternative approach to reduce the computational costs for the calculation of a matrix function φ(L) is to split the graph in smaller subgraphs, using for instance metric clustering techniques as J-center clustering [3,4] or hierarchical partitioning trees [11]. The single domains of a partitioning are then enlarged to create an overlapping cover of the graph.…”

“…The main idea of this approach is to calculate the elements of φ(L) locally on the single subdomains, and then to use a partition of unity to glue the components together. In [3], this approach was investigated and particularly for the variational spline it turned out that with increasing overlapping of the domains the merged local kernels converged rapidly towards the global one. Nevertheless, the block Krylov methods studied in this article can also be used as subroutines in [3,4] to speed up the local GBF calculations.…”

“…In [3], this approach was investigated and particularly for the variational spline it turned out that with increasing overlapping of the domains the merged local kernels converged rapidly towards the global one. Nevertheless, the block Krylov methods studied in this article can also be used as subroutines in [3,4] to speed up the local GBF calculations. (ii) Kernels on graphs are not only relevant for regression or classification purposes in kernel machines.…”