When AI Meets Topology—Why Collaborative Filtering and Contextual Word Embeddings Fail

Collaborative Filtering algorithm requires computation of similarities. By exploring the similarity structure using vector field-based topology, we prove that the algorithm is invalid. Similarly, we can apply an analogous approach to the Contextual Word Embedding algorithms. By reducing the similarity structures into transverse fields which can be decomposed into tangential vectors and normal vectors of the underlying topological structures, we utilize the Poincare-Hopf Index Theorem to prove that the shape of contextual word embedding vectors is not a manifold.