A neighborhood-preserving translation operator on graphs

Abstract

In this paper, we introduce translation operators on graphs. Contrary to spectrally-defined translations in the framework of graph signal processing, our operators mimic neighborhood-preserving properties of translation operators defined in Euclidean spaces directly in the vertex domain, and therefore do not deform a signal as it is translated. We show that in the case of grid graphs built on top of a metric space, these operators exactly match underlying Euclidean translations, suggesting that they completely leverage the underlying metric. More generally, these translations are defined on any graph, and can therefore be used to process signals on those graphs. We show that identifying proposed translations is in general an NP-Complete problem. To cope with this issue, we introduce relaxed versions of these operators, and illustrate translation of signals on random graphs.

Atomic Filter: a Weak Form of Shift Operator for Graph Signals

“…To process such signals, one needs to extend the well-developed theory of classical signal processing to graph signals. There have been a lot of researches on graph signal processing, including graph shift operators [4,5,6,7,8,9,10,11,12,13], graph filters [14,15,16,17,18], graph Fourier transforms [19,20,21], windowed graph Fourier transforms [4,22], graph wavelets [23,24,25,26,27], graph signal sampling [28,29,30,31,32,33,34,35,36,37], multiscale analysis [38,39,40], and approximation theory for graph signals [41,42,43].…”

“…Later in [9], A. Gavili et al defined a set of norm-preserving graph shift operators flexible to accommodate desired properties. In [10,11], N. Grelier et al proposed a definition relying on neighborhood preserving properties. In [12], B. S. Dees et al employed the maximum entropy principle to define a general class of shift operators for random signals on a graph.…”