LightConeFM: Unconstrained Lorentz Embeddings for Collaborative Filtering

Abstract

Hyperbolic embedding methods for collaborative filtering constrain all representations to the hyperboloid manifold, imposing a single geometry regardless of data characteristics. We introduce LightConeFM, which removes this constraint and allows embeddings to freely occupy any causal region of Lorentz-Minkowski space-timelike, lightlike, or spacelike-using only standard gradient descent without Riemannian optimization. Experiments on four real-world datasets reveal two consistent findings: (1) unconstrained embeddings outperform their constrained counterparts on every dataset (up to +7.0% AUC), and (2) the learned causal zone distribution predicts where hyperbolic geometry provides benefit over Euclidean alternatives (Pearson $r=0.94$, Spearman $\rho=1.0)$ -datasets with predominantly timelike users exhibit the largest gains, while predominantly spacelike datasets are better served by Euclidean methods. On a job recommendation dataset, LightConeFM achieves +5.99% AUC improvement in cold-start settings, indicating particular value for sparse, hierarchical domains.