Directed Metric Spaces, Alcoved Polytopes and Large Language Models

Explore a mathematical lecture that delves into the fundamental structures underlying Large Language Models through the lens of directed metric spaces and alcoved polytopes. Learn how conditional probability distributions in text generation can be reframed using -log probabilities to create a directed metric structure on the space of texts. Discover the properties of P(L), a directed metric alcoved polytope containing isometrically embedded texts as generators of extremal rays, and understand how this space encodes semantic information about language. Examine the duality theorem connecting text extensions and restrictions, while gaining insights into the categorical interpretation of these structures, including the Yoneda embedding and its relationship to traditional views of language as monoids or posets. Follow along as the speaker presents collaborative research with Stéphane Gaubert, offering a novel mathematical framework for understanding the architecture and behavior of Large Language Models.