Infinity Embeddings: Representation learning with ultrametric structure for faster search via Constrained Learning.

Learned vector representations have been mostly optimized to capture semantic fidelity while vector indexing and search algorithms typically focus on improving speed, yet these have evolved mostly independently. We present Infinity Embeddings, a constrained learning approach that jointly optimizes for semantic similarity while imposing geometric structure conducive to efficient search. Explicitly, we encourage an ultrametric structure that, under strict satisfaction, yields logarithmic worst-case search complexity. The proposed vector representation is trained with sample-wise constraints that preserve local neighbor structure while enforcing geometry that enables pruning during nearest-neighbor search. We empirically validate our approach on multi-modal vector retrieval, reporting both embedding semantic quality and vector-search complexity measured by the number of comparisons.