Universal Latent Homeomorphic Manifolds: Cross-Domain Representation Learning via Homeomorphism Verification

We present the Universal Latent Homeomorphic Manifold (ULHM), a framework that unifies semantic representations (e.g., human descriptions, diagnostic labels) and observation-driven machine representations (e.g., pixel intensities, sensor readings) into a single latent structure. Despite originating from fundamentally different pathways, both modalities capture the same underlying reality. We establish \emph{homeomorphism}, a continuous bijection preserving topological structure, as the mathematical criterion for determining when latent manifolds induced by different semantic-observation pairs can be rigorously unified. This criterion provides theoretical guarantees for three critical applications: (1) semantic-guided sparse recovery from incomplete observations, (2) cross-domain transfer learning with verified structural compatibility, and (3) zero-shot compositional learning via valid transfer from semantic to observation space. Our framework learns continuous manifold-to-manifold transformations through conditional variational inference, avoiding brittle point-to-point mappings. We develop practical verification algorithms, including trust, continuity, and Wasserstein distance metrics, that empirically validate homeomorphic structure from finite samples. Experiments demonstrate: (1) sparse image recovery from 5\% of CelebA pixels and MNIST digit reconstruction at multiple sparsity levels, (2) cross-domain classifier transfer achieving 86.73\% accuracy from MNIST to Fashion-MNIST without retraining, and (3) zero-shot classification on unseen classes achieving 89.47\% on MNIST, 84.70\% on Fashion-MNIST, and 78.76\% on CIFAR-10. Critically, the homeomorphism criterion correctly rejects incompatible datasets, preventing invalid unification and providing a feasible way to principled decomposition of general foundation models into verified domain-specific components.