Explain the robustness phase transition between two- and four-dimensional oscillators

Characterize why the two-dimensional oscillator regime in Oscillatory Predictive Learning appears robust whereas increasing the oscillator dimension to four causes robust accuracy to collapse, and determine whether the hypothesized phase transition arises because four-dimensional oscillators can embed adversarial perturbations orthogonally to the consensus manifold.

Background

The paper reports that Oscillatory Predictive Learning (OPL) is robust in the N=2N=2 oscillator regime but loses nearly all adversarial robustness when the oscillator dimension increases to N=4N=4, even though clean accuracy improves. The authors hypothesize that two-dimensional oscillators enforce strict degree-1 phase consensus, while four-dimensional oscillators provide additional degrees of freedom that may allow adversarial perturbations to bypass the structural defense.

This conjecture is presented as a mechanistic explanation for the observed non-monotonic relationship between clean accuracy and adversarial robustness. Resolving it would clarify whether the phenomenon is specific to the AKOrN architecture or reflects a broader robustness phase transition in low-dimensional couplings and other iterative dynamical models.

References

We conjecture that the $N{=}2$ regime enforces strict degree-1 phase consensus (equivalent to a 2D rotation), creating a tightly coupled dynamical system in which individual features cannot independently align with adversarial gradient vectors.

— Neither Adversarial Training Nor Purification: Emergent Adversarial Robustness from Oscillatory Predictive Learning  (2609.08683 - Habibi et al., 8 Sep 2026) in Section "Discussion, Trade-offs, and Limitations," paragraph "Mechanistic hypotheses"