A Controlled Evaluation of Curvature Inspired Scalar Feature Scaling for Domain Generalization

1 Introduction

• HyperDG scales Euclidean features by learned positive source-slot scalars during training, then predicts with unscaled Euclidean features and a linear classifier.The paper says this does not mean the representation learned sectional curvature.Source: §1 · p. 1
• The study asks whether the complete package improves held-out-domain accuracy over native DomainBed ERM and whether separate learned source scales offer a practical macro advantage over one shared scale.The second comparison isolates source-scale parameterization more cleanly; neither condition uses an operative between-scale dispersion penalty.Source: §1 · p. 2
• The implementation audit separates its geometric motivation from the executed computation, including an inactive local-gradient term, a scale-cancelling proxy, and a train–inference scaling mismatch.The audit makes the tested intervention precise rather than treating it as interchangeable with manifold-valued learning.Source: §1 · pp. 1–2
• The evaluation completes 384 runs across six conditions, four datasets, four held-out targets, and four seeds, with checkpoints selected using source-only validation.Target-clustered uncertainty is used, with each named target and run seed reported separately.Source: §1 · p. 2
• The full package shows no advantage over ERM, while separate source scales are macro-equivalent to one shared scale within the stated margin.These are bounded conclusions under the tested protocol.Source: §1 · p. 2

2 Related Work and Scope Boundary

• The study uses unmodified DomainBed ERM as its comparison and keeps target-domain data out of checkpoint scoring.DomainBed showed that controlled model selection and protocol matching can erase many claimed DG gains (Gulrajani & Lopez-Paz, 2021).Source: §2 · p. 2
• Manifold-valued hyperbolic models use stronger geometric parameterizations than the scalar rescaling audited here.Examples include manifold operations and classifiers, Lorentz-hyperboloid representations, and curvature-learning networks (Ganea et al., 2018; Desai et al., 2023; Fan et al., 2025).Source: §2 · p. 2
• Published hyperbolic-method numbers are not matched controls because those methods use different algorithms, tasks, architectures, and selection protocols.A controlled comparison would require a manifold-valued encoder or head in the same 384-cell protocol.Source: §2 · p. 2

3.1 Executed feature scaling

• Per-domain scales correspond to three source-loader slots whose domain mapping changes with the held-out target; no parameter is learned for the unseen target.The learned scales start from the same initialization as the fixed scales.Source: §3.1 · p. 3
• With 32 examples in each source minibatch, the alignment term weights all three source means equally and pulls each toward their unweighted mean.Each domain mean is computed from scaled features.Source: §3.1 · p. 3

3.2 Intended versus executed auxiliary terms

• The norm/sharpness proxy collapses to raw feature-norm dispersion, so its value does not depend on the scale and it is not curvature-sensitive or SAM.The implementation computes population variance within each source minibatch before averaging the three domain values.Source: §3.2 · pp. 3–4
• The intended local scale-gradient term is identically zero in primary runs because automatic differentiation returns None, which the implementation converts to zero.The executed objective includes the local term, but its contribution is zero.Source: §3.2 · p. 4

3.3 Six controlled conditions

• The primary contrasts are FullPerDomain versus StandardERM for package utility and NoVariance versus SharedLearnable for domain specificity; MatchedControl and Frozen are descriptive diagnostics.Frozen changes learnability, so it does not substitute for the shared-scale comparison.Source: §3.3 · p. 4, Table 2 · p. 4

4 Experimental Design

• PACS, VLCS, OfficeHome, and TerraIncognita cover shifts in depiction, acquisition, object context, and camera location.The suite-level estimand is the equal-weighted macro over these four benchmarks, not a broader population of DG tasks.Source: §4 · pp. 4–5
• Each environment is held out once, with the other three as sources; all methods use a pretrained ResNet-50, matched splits, and source-only model selection.Seeds vary data splits, initialization, minibatch order, and augmentation; training uses 5,000 updates.Source: §4 · p. 5
• HyperDG makes six training-mode BatchNorm updates per step, versus one for native ERM, so their comparison evaluates the full implementation rather than curvature alone.The shared-update-route NoVariance–SharedLearnable comparison cancels this BatchNorm asymmetry.Source: §4 · p. 5
• Checkpoint selection uses only source environments; held-out-target accuracy is read after the selection rule fixes the checkpoint.Exact ties choose the earliest checkpoint; target curves were logged but are not used by the frozen selection rule.Source: §4 · p. 5
• Seeds are averaged within each target, and the four target effects are the per-dataset inferential units; the exact sign-flip test assumes exchangeable target-level sign flips under the joint no-effect null.The four dataset effects are the clusters; the smallest possible two-sided exact p-value is .125.Source: §4 · pp. 5–6
• Practical equivalence uses a revision-stage ±1 pp SESOI and is scoped to the fixed four-dataset suite macro.TOST and the t3 interval are secondary working-model summaries, not substitutes for the primary Holm-corrected exact test.Source: §4 · p. 6

5.1 Completed six-condition study

• All 384 planned runs completed, and Table 3 reports equal-target macro accuracies, while the cited passage places named targets, seeds, and between-target and within-target run-seed variability in the supplement.Source: §5.1 · p. 6

5.2 Primary utility and domain-specificity contrasts

• −1.01, −0.77, −1.69, and −2.96 pp favor StandardERM over FullPerDomain on PACS, VLCS, OfficeHome, and TerraIncognita, but Holm-adjusted p = .250 provides no adjusted evidence of a difference.The exact four-dataset sign-flip test gives p = .125; the primary Holm-adjusted result governs the claim rather than the secondary t3 interval.Source: §5.2 · p. 6
• −0.13 pp is the NoVariance−SharedLearnable suite effect, with 90% CI [−0.43, 0.17] inside ±1 pp (TOST p = .003), supporting macro-equivalence at the source-selected endpoint.The conclusion is limited to the equal-dataset/equal-target macro; individual dataset and target intervals are wider.Source: §5.2 · p. 7
• Source selection improves over update 4999 in 20 of 24 condition-by-dataset cells by 0.97 pp on average, but update-4999 equivalence is not established.At update 4999, NoVariance−SharedLearnable is −0.32 pp with 90% CI [−1.13, 0.50] and TOST p = .072; equivalence is restricted to the source-selected endpoint.Source: §5.2 · p. 7
• The local-gradient regularizer is zero in all primary logs, and the norm proxy cannot identify curvature, leaving the close scale-condition performance consistent with a weakly identified training-time scale.This result is not evidence about how a correctly implemented manifold-valued curvature parameter would behave.Source: §5.2 · pp. 7–8

6 Discussion

• The complete implementation provides no evidence of improvement over native ERM under the tested protocol, while the separate-scale contrast is equivalent only for the source-selected macro endpoint.Frozen changes scale learnability and is not a substitute for a shared-learnable condition.Source: §6 · p. 8
• The four benchmarks span artistic style, dataset acquisition, object context, and camera-location shifts, but provide no ground-truth geometry for validating learned scalars as curvature.Without known hierarchies, distances, or target curvatures, differing scales alone cannot establish physical or sectional curvature.Source: §6 · p. 8
• A genuine follow-up would retain the present splits, seeds, backbone, budget, and source-only selector while using a correctly implemented Poincaré or Lorentz representation and hyperbolic classifier.Fixed- versus learned-curvature versions would separate manifold geometry from scalar flexibility; incomplete MERU runs or post-hoc Poincaré projections would not answer that question.Source: §6 · p. 8

7 Limitations and Broader Impact

• Only four target clusters per benchmark and four dataset clusters at suite level make exact tests coarse and per-target intervals potentially wide.The study also uses one ImageNet-pretrained ResNet-50 recipe and four classification benchmarks; its SESOI and source-selected endpoint were fixed during revision, not preregistered before the original study.Source: §7 · p. 8
• The implementation has an inactive local term, a scale-cancelling proxy, and unscaled inference, and the study includes no genuine manifold-valued training.These limitations narrow the claim but do not invalidate the negative result for the executed package.Source: §7 · p. 8
• The work uses standard image benchmarks and introduces no new human-subject data; its main impact is methodological, cautioning against treating scalar Euclidean scaling as evidence about hyperbolic learning as a whole.The results should not be cited to dismiss genuine hyperbolic representations.Source: §7 · p. 8

• Across 384 completed runs, the full implementation showed no advantage over native ERM, and all four dataset point estimates were lower.Source: §8 · p. 8
• Separate source-loader scales were equivalent to one shared learned scale within a revision-stage ±1 pp margin for the source-selected suite macro.This equivalence does not extend to every dataset, target, or final-step endpoint.Source: §8 · p. 8
• The negative finding applies to training-time Euclidean rescaling and should not be generalized to manifold-valued hyperbolic models.Source: §8 · p. 8