Angular Gradient Sign Method: Uncovering Vulnerabilities in Hyperbolic Networks

TL;DR

This paper introduces a geometry-aware adversarial attack method for hyperbolic neural networks that focuses on angular perturbations, revealing vulnerabilities in hyperbolic embeddings more effectively than traditional Euclidean-based attacks.

Contribution

It proposes a novel attack leveraging hyperbolic geometry by decomposing gradients into radial and angular components, and applying perturbations in the angular direction to improve attack success.

Findings

Achieves higher fooling rates than conventional methods.

Effectively uncovers vulnerabilities in hyperbolic embeddings.

Provides insights into geometry-specific adversarial strategies.

Abstract

Adversarial examples in neural networks have been extensively studied in Euclidean geometry, but recent advances in \textit{hyperbolic networks} call for a reevaluation of attack strategies in non-Euclidean geometries. Existing methods such as FGSM and PGD apply perturbations without regard to the underlying hyperbolic structure, potentially leading to inefficient or geometrically inconsistent attacks. In this work, we propose a novel adversarial attack that explicitly leverages the geometric properties of hyperbolic space. Specifically, we compute the gradient of the loss function in the tangent space of hyperbolic space, decompose it into a radial (depth) component and an angular (semantic) component, and apply perturbation derived solely from the angular direction. Our method generates adversarial examples by focusing perturbations in semantically sensitive directions encoded in angular movement within the hyperbolic geometry. Empirical results on image classification, cross-modal retrieval tasks and network architectures demonstrate that our attack achieves higher fooling rates than conventional adversarial attacks, while producing high-impact perturbations with deeper insights into vulnerabilities of hyperbolic embeddings. This work highlights the importance of geometry-aware adversarial strategies in curved representation spaces and provides a principled framework for attacking hierarchical embeddings.