dorsal/arxiv
View SchemaUniversal Latent Homeomorphic Manifolds: Cross-Domain Representation Learning via Homeomorphism Verification
| Authors | Tong Wu, Tayab Uddin Wara, Daniel Hernandez, Sidong Lei |
|---|---|
| Categories | |
| ArXiv ID | 2601.09025vv1 |
| URL | https://arxiv.org/abs/2601.09025 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
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. 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.
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"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. 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.",
"arxiv_id": "2601.09025",
"authors": [
"Tong Wu",
"Tayab Uddin Wara",
"Daniel Hernandez",
"Sidong Lei"
],
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"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Universal Latent Homeomorphic Manifolds: Cross-Domain Representation Learning via Homeomorphism Verification",
"url": "https://arxiv.org/abs/2601.09025",
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