Abstract

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 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. When this homeomorphic criterion is satisfied, it enables three critical applications: (1) semantic-guided sparse recovery from incomplete observations, (2) cross-domain transfer learning with empirically assessed structural compatibility, and (3) transductive zero-shot compositional learning via valid transfer from semantic to observation space. Our framework learns continuous manifold-to-manifold transformations through conditional variational inference, with training objectives explicitly designed to enforce bi-Lipschitz homeomorphic properties. We develop practical verification algorithms, including trust, continuity, and Wasserstein distance metrics, that empirically indicate whether the learned representations exhibit properties consistent with homeomorphic structure from finite samples. Experiments demonstrate substantial improvements over state-of-the-art (SOTA) baselines: (1) sparse recovery from 8% of pixels with much lower MSE than SOTA on CelebA under noise, (2) cross-domain transfer achieving 86.73% MNIST$\rightarrow$Fashion-MNIST accuracy without retraining, and (3) transductive zero-shot classification achieving 78.76% on CIFAR-10, exceeding prior work by 16.66%. Critically, the homeomorphism criterion determines when different semantic-observation pairs share compatible latent structure, enabling principled unification into shared representations within the tested domains and suggesting a structured basis for decomposing broad models into domain-specific components.