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A central challenge in source-free domain adaptation (SFDA) is the lack of a theoretical framework for explicitly analyzing domain shifts, as the absence of source data prevents direct domain comparisons. In this paper, we introduce the Vicinal Gaussian Transform (VGT), an analytical operator that models source-informed latent vicinities as Gaussians and shows that vicinal prediction divergence is bounded by their covariance. By this formulation, SFDA can be reframed as shrinking covariance to reinforce label consistency. To operationalize this idea, we introduce the Energy-based VGT (EBVGT), a novel SDE that realizes the Gaussian transform by contracting covariance through a denoising mechanism. A recovery-likelihood with a Schrödinger-Bridge smoothness penalty denoises perturbed states, while a BYOL-derived energy function, directly obtained from model predictions, provides the score to guide label-consistent trajectories within the vicinity. This design not only yields noise-suppressed vicinal features for adaptation without source data, but also eliminates the need for additional learnable parameters for score estimation, in contrast to conventional deep SDEs. Our EBVGT is model- and modality-agnostic, efficient for classification, and improves state-of-the-art SFDA methods by 1.3-3.0% (2.0% on average) across both 2D image and 3D point cloud benchmarks.
Wang et al. (Wed,) studied this question.