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On the Adversarial Robustness of Restricted Hopfield Networks

Year: 2026
Type: article
Abstract: Restricted Hopfield Networks (RHNs) extend classical Hopfield neural networks by incorporating hidden layers and trainable retrieval dynamics, thereby improving storage capacity and stability. This work presents a systematic theoretical and empirical investigation of the adversarial robustness of RHNs. We analyze the Subspace Rotation Algorithm (SRA) and establish two theoretical results: SRA produces optimal semi-orthogonal weight matrices for RHNs, and these weight structures induce non-expansive retrieval dynamics that prevent adversarial perturbations from being amplified. We further identify two factors associated with robustness: the filtering effect of the learned signal subspace and gradient-pattern alignment (GPA). Our analysis indicates that semi-orthogonality and signal-subspace dimensionality play particularly important roles, whereas GPA serves as a complementary robustness indicator. We compare RHNs with Modern Hopfield Networks (MHNs) and Predictive Coding Networks (PCNs) on the Alphabet and Character datasets. Robustness is evaluated under masking, Gaussian noise, and transferred adversarial attacks generated by an independently trained autoencoder, including FGSM, FFGSM, MIFGSM, NIFGSM, BIM, PGD, EOTPGD, Jitter, and SPSA. Quantitative and qualitative results show that RHNs achieve the most consistent retrieval performance across corruption types and perturbation magnitudes. RHNs generally outperform MHNs and remain competitive with or superior to PCNs under transferred adversarial attacks. These results demonstrate that semi-orthogonal weight structures and limited signal subspaces provide RHNs with an effective and theoretically grounded defense against adversarial perturbations.
DOI: 10.1109/tbdata.2026.3728235
Language: English
Open Access status: hybrid
Estimated APC paid: $2,645
OpenAlex ID: W7204490644
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