[Paper Review] Translations on graphs with neighborhood preservation.

Bastien Pasdeloup, Vincent Gripon+3 more|arXiv (Cornell University)|Sep 12, 2017

Advanced Graph Neural Networks13 references3 citations

TL;DR

This paper proposes a novel graph translation definition based on neighborhood preservation, ensuring localized translations and consistent observer-point-of-view behavior. It proves the method reduces to standard geometric translation on regular grids, is robust to edge noise, and identifies translations via a relaxed proxy problem, despite NP-completeness of exact identification.

ABSTRACT

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.

Motivation & Objective

• To define graph translation operators that preserve localization and mimic observer-point-of-view shifts, addressing limitations in prior methods.
• To ensure graph translations align with underlying geometry in regular structures like grid graphs.
• To develop a tractable proxy for identifying translations, given the NP-completeness of exact identification.
• To evaluate robustness of the method under small edge perturbations in noisy graphs.
• To explore the feasibility of translation identification in randomly generated graphs.

Proposed method

• Defines graph translation via neighborhood preservation, ensuring the local structure around a vertex is maintained under translation.
• Uses a relaxed optimization problem as a proxy to identify valid translations, avoiding the NP-complete exact search.
• Applies the method to regular graphs (e.g., grid graphs) to derive closed-form solutions.
• Employs spectral and structural analysis to validate that translations match standard geometric translations on regular lattices.
• Tests robustness by introducing small edge variations and measuring translation consistency.
• Extends analysis to randomly generated graphs to assess generalizability of the method.

Experimental results

Research questions

• RQ1 Can graph translation be defined such that localized signals remain localized after translation?
• RQ2 Does the proposed method preserve the observer-point-of-view equivalence in regular graphs?
• RQ3 How can translation identification be approximated efficiently given its NP-completeness?
• RQ4 How robust is the method to small edge perturbations in graph structure?
• RQ5 Can the method be applied meaningfully to randomly generated graphs?

Key findings

• The proposed neighborhood-preserving translation definition satisfies both localization and observer-point-of-view consistency, resolving limitations of prior approaches.
• On regular grid graphs, the method reduces to standard geometric translation, confirming its geometric fidelity.
• The method remains robust under small edge variations, maintaining accurate translation identification in noisy graphs.
• A relaxed proxy problem enables efficient identification of translations, making the method scalable despite NP-completeness of exact identification.
• Closed-form solutions are derived for highly regular graphs, validating theoretical consistency.
• Preliminary results suggest feasibility of translation identification in randomly generated graphs, though with reduced accuracy compared to regular structures.