[R] Measuring the Symmetry--Data Exchange Rate

The prediction that equivariance reduces sample complexity by a factor of |G| appears in roughly every paper on geometric deep learning and is measured as an actual scaling law in roughly none of them. This paper does the measurement.

The methodology is the interesting part. Naive estimators conflate group order with task difficulty (larger groups induce harder symmetry structure, not just more constraint), so the authors derive a relative exchange rate that cancels the shared difficulty out, meaning roughly how much less data the equivariant model needs compared to a vanilla baseline as a function of n, on a controlled C_n-symmetric task where n is a free knob. They also pre-specify a failure taxonomy: explicit conditions that would count as evidence against the hypothesis before seeing results.

The headline number is beta_diff ~ 1.28, consistent with the theoretical 1.0. But the more durable finding is the wrong-group control: a model built with the wrong cyclic symmetry, same orbit size and same compute budget, is actively worse than no constraint. Not noise. The joint pairwise CI [+0.79, +3.26] excludes zero robustly across every estimator they run. Misalignment isn't just unhelpful; it is harmful.

There is also a clean mathematical result slipped into Sec. 4.3: augmentation + test-time orbit averaging is exactly equivariant for output-pooling architectures, provably and verified to bit-identical training curves. The architecture-vs-augmentation gap collapses to whether you apply the orbit average at test time, not to anything structural. This seems underappreciated.

The paper is unusually transparent about what it didn't nail: the relative-rate estimator was adopted post-hoc, the two-level bootstrap CI (seeds x group sizes) includes zero, and a finer-N replication on a sqrt(2)-spaced grid is inconclusive. They rank their findings explicitly by robustness. The wrong-group result is the one they would stake a claim on. The exchange rate is directionally probable.