Методы геометрии и топологии для исследования моделей глубокого обученияMethods of geometry and topology in the study of deep learning models

This dissertation is devoted to the development and application of geometric and topological methods for analyzing, interpreting, evaluating, and improving deep learning models. The work investigates geometric and topological properties of internal data representations (embeddings) in various Deep Neural Network (DNN) architectures, including convolutional DNNs and Transformers under different training modes (e.g. supervised, self-supervised learning etc). A method for evaluating model performance based on geometric properties of intermediate representations is proposed. The research develops an approach for improving model performance through integration of domain-specific topological data information into embeddings and proposes a modification of the RNN architecture called TonnetzNet, demonstrating performance enhancement. The work also investigates the applicability of the fractal intrinsic dimension of Transformer embedding data manifold for robust detection of content generated by large language models (LLMs) and presents a method for synthetic image recognition. Finally, the capabilities and limitations of various language models are evaluated in tasks of modeling symbolic sequences representing elements from normal closures in free groups. The dissertation provides both theoretical insights and empirical validation of the developed methods. The research results contribute to the fields of AI safety, mechanistic interpretability, and explainable AI. Overall, the work emphasizes the promise and importance of geometric and topological methods not only as analytical tools, but also as a foundation for developing approaches to interpretation, evaluation, application, and improvement of DNNs.