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While in-context learning (ICL) has achieved remarkable success in natural language and vision domains, its theoretical understanding-particularly in the context of structured geometric data-remains unexplored. This paper initiates a theoretical study of ICL for regression of H\"older functions on manifolds. We establish a novel connection between the attention mechanism and classical kernel methods, demonstrating that transformers effectively perform kernel-based prediction at a new query through its interaction with the prompt. This connection is validated by numerical experiments, revealing that the learned query-prompt scores for H\"older functions are highly correlated with the Gaussian kernel. Building on this insight, we derive generalization error bounds in terms of the prompt length and the number of training tasks. When a sufficient number of training tasks are observed, transformers give rise to the minimax regression rate of H\"older functions on manifolds, which scales exponentially with the intrinsic dimension of the manifold, rather than the ambient space dimension. Our result also characterizes how the generalization error scales with the number of training tasks, shedding light on the complexity of transformers as in-context kernel algorithm learners. Our findings provide foundational insights into the role of geometry in ICL and novels tools to study ICL of nonlinear models. 

Not AvailableReliable machine learning and statistical analysis rely on diverse, well-distributed training data. However, real-world datasets are often limited in size and exhibit underrepresentation across key subpopulations, leading to biased predictions and reduced performance, particularly in supervised tasks such as classification. To address these challenges, we propose Conditional Data Synthesis Augmentation (CoDSA), a novel framework that leverages generative models, such as diffusion models, to synthesize high-fidelity data for improving model performance across multimodal domains, including tabular, textual, and image data. CoDSA generates synthetic samples that faithfully capture the conditional distributions of the original data, with a focus on under-sampled or high-interest regions. Through transfer learning, CoDSA fine-tunes pre-trained generative models to enhance the realism of synthetic data and increase sample density in sparse areas. This process preserves inter-modal relationships, mitigates data imbalance, improves domain adaptation, and boosts generalization. We also introduce a theoretical framework that quantifies the statistical accuracy improvements enabled by CoDSA as a function of synthetic sample volume and targeted region allocation, providing formal guarantees of its effectiveness. Extensive experiments demonstrate that CoDSA consistently outperforms non-adaptive augmentation strategies and state-of-the-art baselines in both supervised and unsupervised settings. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.