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

Sunny Gupta; Amit Sethi

cs.LGOct 13, 2026

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

Authors: Sunny Gupta, Amit Sethi

Organizations: Indian Institute of Technology Bombay Mumbai, Maharashtra, India

Abstract

Domain-specific curvature is an appealing idea for domain generalization (DG), but curvature learning can denote substantially different computations. This paper evaluates one deliberately narrow implementation: the code calls it HyperDG, but it rescales Euclidean encoder features during training by learned positive scalars and uses an ordinary linear classifier on unscaled Euclidean features at inference. It does not implement a Lorentz or Poincaré representation, geodesic prediction, or a Sharpness-Aware Minimization inner loop. We audit the executed objective and evaluate six conditions on PACS, VLCS, Office Home, and TerraIncognita. The completed design contains 384 runs: six conditions, four held-out targets, and four run seeds per dataset. Checkpoints are selected using only the three source-validation environments. Inference first averages paired seed differences within each held-out target; targets, rather than 16 seed-by-target observations, are the per-dataset clusters. The full implementation's equal-dataset macro point estimate is 1.61 percentage points below native DomainBed ERM (95% clustered CI [-3.17, -0.04]). All four dataset point estimates favor ERM, but an exact four-dataset sign-flip test gives p = .125 and Holm-adjusted p = .250; we therefore claim no adjusted evidence of a difference. The cleaner domain-specificity contrast, separate learned source scales without their variance penalty versus one shared learned source scale, is -0.13 pp (95% CI [-0.53, 0.27]). Its 90% CI lies within a revision-stage $\pm 1$ pp smallest effect of practical interest (TOST p = .003), supporting equivalence only for the source-selected suite macro, not every dataset, target, or final-step endpoint. The study thus supplies a bounded negative result for this scalar training package; it does not test or refute genuine hyperbolic representation learning.

Figures & tables

Family	Representation and prediction	Relation to this study
Hyperbolic neural networks ( Ganea et al., 2018 )	Poincaré/Möbius operations and a hyperbolic classifier	Genuine manifold baseline; not implemented here
MERU and learned-curvature HNNs ( Desai et al., 2023 ; Fan et al., 2025 )	Lorentz or hyperbolic embeddings with explicit manifold geometry	Stronger geometric parameterization; different protocol
Hyperbolic DG ( Bi et al., 2025 ; Di Salvo et al., 2026 )	Hyperbolic representations embedded in DG-specific objectives	Relevant recent direction; not a scalar-only ablation
HyperDG (this audit)	$h_{\theta}(x)/\sqrt{c_{i}}$ in training; Euclidean linear prediction on $h_{\theta}(x)$	Curvature-inspired scalar feature scaling only

Table 1: The claim boundary. The last row is the only mechanism tested in this paper. Hyperbolic in the other rows denotes manifold-valued operations or representations, not merely a scalar applied to Euclidean features.

Condition	Executed treatment
StandardERM	Native DomainBed ERM; no scales or auxiliary losses.
MatchedControl	HyperDG route; three fixed local scales and one fixed global scale; all five coefficients zero.
FullPerDomain	Three learned local scales and one learned global scale; full coefficients in Equation 8 .
Frozen	All scales fixed; retains alignment, norm, local (inactive), and global terms; $\alpha=0$ .
NoVariance	Three learned local scales; same remaining objective as FullPerDomain but $\alpha=0$ .
SharedLearnable	One learned local scale shared by all three source loaders plus the separate learned global scale. Scale dispersion is structurally zero.

Table 2: Completed conditions. All five non-ERM conditions use the same HyperDG update route. Only StandardERM , FullPerDomain , NoVariance , and SharedLearnable enter the paper’s two primary contrasts; MatchedControl and Frozen are descriptive diagnostics (see text).

Condition	PACS	VLCS	OfficeHome	TerraInc.
StandardERM	85.88	77.19	70.21	47.13
MatchedControl	84.22	75.38	70.05	44.14
FullPerDomain	84.87	76.42	68.52	44.17
Frozen	86.14	75.84	67.81	44.20
NoVariance	85.28	76.17	68.36	43.70
SharedLearnable	85.55	76.56	68.17	43.75

Table 3: Source-validation-selected target accuracy (%). Each cell averages four named targets after averaging four run seeds within target. Bold marks the highest point estimate in a dataset, not a significance claim.

Contrast	Effect [95% CI]	Exact / Holm $p$	TOST $p$	Interpretation
Full $-$ ERM	$-1.61$ $[-3.17,-0.04]$	.125 / .250	.848	No adjusted evidence; equivalence not demonstrated
NoVar $-$ Shared	$-0.13$ $[-0.53,0.27]$	.375 / .375	.003	$\pm 1\,pp$ source-selected macro-equivalent

Table 4: Primary suite-macro contrasts. Exact

p

-values sign-flip the four dataset effects; Holm adjusts the two primary tests. TOST uses the

\pm 1\,pp

revision-stage SESOI.

Figure 1: Paired effects for the two primary contrasts. Blue intervals are

t_{3}

intervals across four paired run seeds within a named target; orange intervals cluster four target means within a dataset; black intervals cluster the four fixed dataset means. The visible target heterogeneity is why seed-by-target cells are not treated as 16 independent domain draws. These intervals describe this fixed, heterogeneous benchmark suite, not all DG tasks. Exact values and run-seed SDs are in the supplement.

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