Title: Quantifying the Symmetry–Data Exchange Rate

Abstract

While equivariance theory posits that imposing architectural symmetry priors can lower sample complexity by a factor of |G|, this claim is frequently cited yet seldom validated as a scaling law with rigorous controls to isolate the prior from confounding variables. In this study, we examine a controlled C_n-symmetric task and present three key observations.

First, we find that employing a "wrong-group" control—defined by an identical orbit size and matched computational resources—yields performance inferior to having no constraints whatsoever. The joint pairwise confidence interval [+0.79, +3.26] excludes zero, a result that remains robust across various estimators. This indicates that applying a misaligned constraint is not merely ineffective but actively detrimental.

Second, an augmentation baseline utilizing test-time orbit averaging achieves performance identical to that of the equivariant model. Specifically, validation curves are bit-identical per epoch across matched cells. This demonstrates that the performance gap between architecture and augmentation is conditional upon asymmetric test-time computation, rather than being an unconditional architectural advantage.

Third, the relative exchange rate, denoted as beta_diff = 1.28, aligns in both sign and order of magnitude with the theoretical prediction of 1.0. The single-level confidence interval [+0.92, +2.05] supports this consistency. However, a more conservative two-level bootstrap analysis (varying seeds and group sizes) broadens the interval to [-0.63, +1.72], which includes zero. Furthermore, a high-resolution replication on a grid spaced by sqrt(2) yields an inconclusive point estimate of -0.82.

Methodologically, this work introduces a relative-rate estimator designed to cancel out shared-difficulty confounds, alongside a wrong-group control and a pre-specified failure taxonomy. These tools are applicable to any inductive bias whose strength can be parameterized. We maintain honest scoping regarding our methodology: the primary estimator beta_diff was selected post-hoc after initial analyses identified a positive-slope identifiability issue; the study design was not externally pre-registered; and the primary metric relies on an OLS slope derived from seven group sizes on a coarse N grid. Consequently, this should be viewed as an exploratory study rather than a confirmatory measurement. The finding regarding the wrong-group control represents the cleanest result and is reported with the highest degree of confidence. A registered replication using fresh seeds remains a direction for future work.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC