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

2026-01-13

AI Summary

This study addresses the lack of structurally consistent representations between semantic descriptions and observational data such as images or sensor readings. It proposes a novel framework grounded in the theoretical principle of homeomorphism, unifying the latent manifolds of semantic and observation spaces into a common latent homeomorphic manifold. For the first time, homeomorphism verification is introduced as a theoretical foundation for cross-domain representation learning to ensure manifold structural compatibility. The method employs conditional variational inference to learn continuous manifold-to-manifold mappings and introduces a homeomorphism verification algorithm based on credibility, continuity, and Wasserstein distance. Experiments demonstrate 5% pixel-sparse recovery on CelebA and MNIST, 86.73% accuracy in MNIST→Fashion-MNIST cross-domain transfer, and zero-shot classification performance of 89.47%, 84.70%, and 78.76% on MNIST, Fashion-MNIST, and CIFAR-10, respectively, while effectively rejecting incompatible data.

Technology Category

Machine Learning: Learning with ManifoldsComputer Vision: Representation Learning for VisionApplication Domains: Humanities & Computational Social Science

Application Category

Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsSearch and Retrieval-Augmented AI: Vertical and domain-specific search

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 78.76% on CIFAR-10. 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.

Problem

• cross-domain representation learning
• homeomorphism
• latent manifold
• semantic-observation alignment
• topological structure

Innovation

• homeomorphism
• latent manifold
• cross-domain representation learning
• conditional variational inference
• zero-shot compositional learning