Tianyu Bell Pan

Tianyu Bell Pan, Damon L. Woodard

Large language models (LLMs) demonstrate strong performance, but they often lack transparency. We introduce GeoLAN, a training framework that treats token representations as geometric trajectories and applies stickiness conditions inspired by recent developments related to the Kakeya Conjecture. We have developed two differentiable regularizers, Katz-Tao Convex Wolff (KT-CW) and Katz-Tao Attention (KT-Attn), that promote isotropy and encourage diverse attention. Our experiments with Gemma-3 (1B, 4B, 12B) and Llama-3-8B show that GeoLAN frequently maintains task accuracy while improving geometric metrics and reducing certain fairness biases. These benefits are most significant in mid-sized models. Our findings reveal scale-dependent trade-offs between geometric precision and performance, suggesting that geometry-aware training is a promising approach to enhance mechanistic interpretability.

Tianyu Bell Pan, Mengdi Zhu, Alexa Jordyn Cole et al.

Multimodal learning systems often struggle in non-stationary environments due to concept drift, where changing data distributions can degrade performance. Modality-specific drifts and the lack of mechanisms for continuous, stable adaptation compound this challenge. This paper introduces LS-OGD, a novel adaptive control framework for robust multimodal learning in the presence of concept drift. LS-OGD uses an online controller that dynamically adjusts the model's learning rate and the fusion weights between different data modalities in response to detected drift and evolving prediction errors. We prove that under bounded drift conditions, the LS-OGD system's prediction error is uniformly ultimately bounded and converges to zero if the drift ceases. Additionally, we demonstrate that the adaptive fusion strategy effectively isolates and mitigates the impact of severe modality-specific drift, thereby ensuring system resilience and fault tolerance. These theoretical guarantees establish a principled foundation for developing reliable and continuously adapting multimodal learning systems.

Tianyu Bell Pan, Damon L. Woodard

This paper explores a multimodal co-training framework designed to enhance model generalization in situations where labeled data is limited and distribution shifts occur. We thoroughly examine the theoretical foundations of this framework, deriving conditions under which the use of unlabeled data and the promotion of agreement between classifiers for different modalities lead to significant improvements in generalization. We also present a convergence analysis that confirms the effectiveness of iterative co-training in reducing classification errors. In addition, we establish a novel generalization bound that, for the first time in a multimodal co-training context, decomposes and quantifies the distinct advantages gained from leveraging unlabeled multimodal data, promoting inter-view agreement, and maintaining conditional view independence. Our findings highlight the practical benefits of multimodal co-training as a structured approach to developing data-efficient and robust AI systems that can effectively generalize in dynamic, real-world environments. The theoretical foundations are examined in dialogue with, and in advance of, established co-training principles.