Universal Latent Homeomorphic Manifolds: A Framework for Cross-Domain Representation Unification

TMLR Paper7343 Authors

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 \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. 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 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, 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 validate whether the learned representations achieve 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) 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 universal representations and providing a mathematical foundation for decomposing general foundation models into domain-specific components.

Submission Type: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Gabriel_Loaiza-Ganem1

Submission Number: 7343