Translations on graphs with neighborhood preservation

In the field of graph signal processing, defining translation operators is crucial to allow certain tasks, including moving a filter to a specific location or tracking objects. In order to successfully generalize translation-based tools existing in the time domain, graph based translations should offer multiple properties: a) the translation of a localized kernel should be localized, b) in regular cases, translating a signal to a vertex should have similar effect to moving the observer's point of view to this same vertex. In previous work several definitions have been proposed, but none of them satisfy both a) and b). In this paper we propose to define translations based on neighborhood preservation properties. We show that in the case of a grid graph obtained from regularly sampling a vector space, our proposed definition matches the underlying geometrical translation. We point out that identification of these graph-based translations is NP-complete and propose a relaxed problem as a proxy to find some of them. Our results are illustrated on highly regular graphs on which we can obtain closed form for the proposed translations, as well as on noisy versions of such graphs, emphasizing robustness of the proposed method with respect to small edge variations. Finally, we discuss the identification of translations on randomly generated graph.