Probabilistic Geometric Alignment via Bayesian Latent Transport for Domain-Adaptive Foundation Models

Abstract

Adapting large-scale foundation models to new domains with limited supervision remains a fundamental challenge due to latent distribution mismatch, unstable optimization dynamics, and miscalibrated uncertainty propagation. This paper introduces an uncertainty-aware probabilistic latent transport framework that formulates domain adaptation as a stochastic geometric alignment problem in representation space. A Bayesian transport operator is proposed to redistribute latent probability mass along Wasserstein-type geodesic trajectories, while a PAC-Bayesian regularization mechanism constrains posterior model complexity to mitigate catastrophic overfitting. The proposed formulation yields theoretical guarantees on convergence stability, loss landscape smoothness, and sample efficiency under distributional shift. Empirical analyses demonstrate substantial reduction in latent manifold discrepancy, accelerated transport energy decay, and improved covariance calibration compared with deterministic fine-tuning and adversarial domain adaptation baselines. Furthermore, bounded posterior uncertainty evolution indicates enhanced probabilistic reliability during cross-domain transfer. By establishing a principled connection between stochastic optimal transport geometry and statistical generalization theory, the proposed framework provides new insights into robust adaptation of modern foundation architectures operating in heterogeneous environments. These findings suggest that uncertainty-aware probabilistic alignment constitutes a promising paradigm for reliable transfer learning in next-generation deep representation systems.

Figures (8)

• Figure 1: Probabilistic foundation model adaptation showing latent density mismatch, Bayesian posterior transport alignment, stochastic representation regularization, and PAC-Bayesian risk bound tightening.
• Figure 2: Taxonomy of domain adaptation paradigms highlighting the conceptual positioning of the proposed probabilistic latent transport framework integrating Bayesian modeling and PAC-Bayesian generalization theory.
• Figure 3: Evolution of domain adaptation paradigms illustrating the transition from deterministic feature alignment toward uncertainty-aware probabilistic adaptation for foundation models.
• Figure 4: Convergence behavior of the latent geometry discrepancy metric during domain adaptation. The proposed probabilistic latent transport framework exhibits a significantly faster and more stable reduction in manifold mismatch compared with deterministic fine-tuning and adversarial alignment baselines. The shaded regions indicate variance across multiple training seeds, demonstrating improved optimization robustness and reduced sensitivity to stochastic initialization.
• Figure 5: Evolution of Wasserstein transport energy throughout the adaptation process. A steep monotonic decay is observed for the proposed method, indicating efficient redistribution of latent probability mass along probabilistic transport trajectories. This behavior empirically supports the theoretical formulation of uncertainty-aware geometric alignment and confirms accelerated convergence of the stochastic transport dynamics.