Resolving Homotopy Obstructions in Normalizing Flows via Contractible Covers

Normalizing Flows are a powerful class of likelihood-based generative models that map complex target distributions into simple, tractable base distributions, usually Gaussians. As differentiably invertible functions trained via likelihood maximization, they enable exact density evaluation and efficient sampling. However, they also suffer from the fundamental topological rigidity of homeomorphic transformations: they cannot alter the topology, and consequently homotopy, of the space they transform without severely increasing their bi-Lipschitz constant. $\text{Co}\text{ntractible}\text{Co}\text{ver}\text{Flow}(\text{CoCoFlow})$ resolves these homotopy obstructions by partitioning the target support into a data-driven $\text{contractible cover}$ and then training a posterior-regularized $\text{mixture of Normalizing Flows}$ confined to the prescribed cover elements, which are guaranteed to be homotopically compatible with the Gaussian base by construction. The mixture model preserves exact likelihood evaluation, deterministic maximum likelihood training, and efficient sampling of standard Normalizing Flows. Experiments on synthetic and real-world data distributions demonstrate improved or competitive density estimation with increased continuity of the generative map compared to standard baselines. To our knowledge, CoCoFlow is the first mixture-of-normalizing-flows framework to provide per-component contractibility guarantees through a topology-aware, data-driven partition of the target support, offering a principled resolution to a long-standing structural limitation of the field.