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Angular Gradient Sign Method: Uncovering Vulnerabilities in Hyperbolic Networks

Minsoo Jo, Dongyoon Yang, Taesup Kim

TL;DR

This work evaluates adversarial robustness in hyperbolic neural networks and demonstrates that conventional Euclidean attacks may fail to fully exploit hyperbolic geometry. It introduces the Angular Gradient Sign Method (AGSM), which computes gradients in the tangent space, decomposes perturbations into radial and angular components, and backpropagates only the angular direction to craft perturbations. The approach is extended to Projected Angular Gradient Descent (PAGD) for multi-step attacks, with extensive experiments on image classification (Poincaré ResNet) and cross-modal retrieval (HyCoCLIP) showing superior fooling rates and larger semantic disruption than baselines. The findings emphasize the need for geometry-aware defenses in curved representation spaces and provide a principled framework for probing vulnerabilities of hierarchical hyperbolic embeddings.

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.

Angular Gradient Sign Method: Uncovering Vulnerabilities in Hyperbolic Networks

TL;DR

This work evaluates adversarial robustness in hyperbolic neural networks and demonstrates that conventional Euclidean attacks may fail to fully exploit hyperbolic geometry. It introduces the Angular Gradient Sign Method (AGSM), which computes gradients in the tangent space, decomposes perturbations into radial and angular components, and backpropagates only the angular direction to craft perturbations. The approach is extended to Projected Angular Gradient Descent (PAGD) for multi-step attacks, with extensive experiments on image classification (Poincaré ResNet) and cross-modal retrieval (HyCoCLIP) showing superior fooling rates and larger semantic disruption than baselines. The findings emphasize the need for geometry-aware defenses in curved representation spaces and provide a principled framework for probing vulnerabilities of hierarchical hyperbolic embeddings.

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.

Paper Structure

This paper contains 27 sections, 16 equations, 2 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Hyerpbolic Networks.
  4. Gradient-based Adversarial Attacks.
  5. Preliminaries
  6. Poincaré Ball Model.
  7. Lorentz Model.
  8. Proposed Method
  9. Geometric Decomposition of FGSM Perturbations.
  10. Angular-based Adversarial Perturbation.
  11. Extension to Projected Gradient Descent.
  12. Experiments
  13. Experimental Setup
  14. Datasets.
  15. Models.
  16. ...and 12 more sections

Figures (2)

  • Figure 1: Overview of representation shifts induced by FGSM, AGSM, radial, and angular perturbations. We visualize how FGSM, AGSM, radial, and angular perturbations influence predictions and confidence (MSP; Maximum Softmax Probability). FGSM causes mixed, less semantic shifts, while radial perturbations reduce confidence without changing labels. AGSM amplifies angular deviation, leading to semantically meaningful misclassifications and stronger confidence drops.
  • Figure 2: Qualitative comparison of Image-to-Text retrieval under FGSM, radial shift, angular shift, and AGSM. While the radial shift preserves the correct caption, FGSM and the standard angular shift generate semantically incorrect outputs, and AGSM yields the most misaligned caption.