Shared Gaussianization: What Gaussian Regularizers Certify About Contrastive Learning, and What They Miss

Ruoyu Zhao, Yuting Chen, Jinheng Zhang, Zhehao Zou, Tong Che

What can a distribution-matching regularizer such as SIGReg in LeJEPA certify about contrastive learning? We study shared Gaussianization (SG), a characteristic-function Gaussianity test on the average of two normalized views, scaled by an independent $χ_d$ radius. Because disagreeing views shorten the average, one test detects both misalignment and non-uniformity. SG vanishes exactly at the aligned, uniform minimizers of population InfoNCE, and under equal marginals it bounds the InfoNCE excess by $4\cdot 3^{3/4}β$ times the square root of the SG loss, plus a term linear in the loss. The square-root rate and this dimension-free constant are sharp, and no squared mean-embedding distance on view pairs achieves a faster rate. With an explicit alignment term, a rotation-invariant uniformity test gives a linear bound if and only if its spectrum dominates that of InfoNCE's kernel $e^{βu^\top v}$; SG's own test does, Gaussian kernels $e^{-γ\|u-v\|^2}$ qualify exactly when $γ\ge β/2$, and moment matching never does. Away from the optimum, the objectives differ. Along an isotropic nuisance channel, pure SG lowers its loss by adding per-view nuisance whenever the shared code is…