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

Tong Wu, Tayab Uddin Wara, Daniel Hernandez, Sidong Lei

TL;DR

The paper introduces the Universal Latent Homeomorphic Manifold (ULHM), a framework that unifies semantic descriptions and raw observations into a shared latent space by enforcing a homeomorphism between modality-induced manifolds. It combines asymmetric training (semantic supervision during learning, but not at inference) with a multi-objective loss that includes reconstruction, cross-modal consistency, and topological preservation, and it provides a practical, three-level homeomorphism verification protocol (global, local, semantic) to ensure validity before transfer or zero-shot tasks. The authors demonstrate three core capabilities: (i) semantic-guided sparse recovery from highly undersampled data, (ii) cross-domain classifier transfer with verified structure enabling zero-shot performance, and (iii) zero-shot classification via preserved topology and nearest-centroid decision rules across multiple datasets. Empirical results on MNIST, Fashion-MNIST, CelebA, and CIFAR-10 show strong gains over baselines in sparse recovery, cross-domain transfer, and zero-shot learning, with robust verification metrics (Betti numbers, Wasserstein distance, trust, and alignment). The approach offers principled decomposability of foundation models into domain-specific components with formal guarantees, and it opens avenues for extension to hierarchical, non-Euclidean, and cyber-physical settings.

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.

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

TL;DR

The paper introduces the Universal Latent Homeomorphic Manifold (ULHM), a framework that unifies semantic descriptions and raw observations into a shared latent space by enforcing a homeomorphism between modality-induced manifolds. It combines asymmetric training (semantic supervision during learning, but not at inference) with a multi-objective loss that includes reconstruction, cross-modal consistency, and topological preservation, and it provides a practical, three-level homeomorphism verification protocol (global, local, semantic) to ensure validity before transfer or zero-shot tasks. The authors demonstrate three core capabilities: (i) semantic-guided sparse recovery from highly undersampled data, (ii) cross-domain classifier transfer with verified structure enabling zero-shot performance, and (iii) zero-shot classification via preserved topology and nearest-centroid decision rules across multiple datasets. Empirical results on MNIST, Fashion-MNIST, CelebA, and CIFAR-10 show strong gains over baselines in sparse recovery, cross-domain transfer, and zero-shot learning, with robust verification metrics (Betti numbers, Wasserstein distance, trust, and alignment). The approach offers principled decomposability of foundation models into domain-specific components with formal guarantees, and it opens avenues for extension to hierarchical, non-Euclidean, and cyber-physical settings.

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.
Paper Structure (62 sections, 3 theorems, 33 equations, 5 figures, 7 tables, 1 algorithm)

This paper contains 62 sections, 3 theorems, 33 equations, 5 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Related Work
  4. Universal Latent Manifold
  5. Homeomorphic Learning
  6. Sparse Recovery and Compressed Sensing
  7. Cross-Domain Transfer and Universal Classifiers
  8. Zero-Shot Learning and Compositional Generalization
  9. Contributions
  10. Universal Latent Homeomorphic Manifold Framework
  11. Problem Formulation and Notation
  12. The Role of the Latent Manifold
  13. The Universal Manifold Problem
  14. Notation Summary
  15. Latent Manifold Learning Architecture
  16. ...and 47 more sections

Key Result

Theorem 1

We denote the data manifold in observation space for dataset $k$ as $\mathcal{M}_x^{[k]}$ and the corresponding latent manifold as $\mathcal{M}_z^{[k]}$. Let $\{\mathcal{M}_x^{[k]}\}_{k=1}^K$ be disjoint data manifolds mapped to latent manifolds $\{\mathcal{M}_z^{[k]}\}_{k=1}^K$ by encoders $\{E^{[k

Figures (5)

  • Figure 1: Overview of the proposed Universal Latent Manifold (ULHM) framework. The architecture bridges the Semantic Space$\mathcal{S}$ (utilized during training with class labels, attributes, and descriptions) and the Observation Space$\mathcal{X}$ (comprising images, multi-modal inputs, and sensor data). This integration enables diverse downstream capabilities, including Sparse Image Recovery and Denoising, Cross-Domain Integration and Transfer, and Zero-Shot Learning.
  • Figure 2: Sparse recovery on the MNIST digits. It shows ground truth images, heavily masked 10%--11% inputs, and the successful corresponding reconstructed image with low MSE.
  • Figure 3: Illustrating the sparse CelebA-style image recovery settings with ground truth and random 5% sparsity.
  • Figure 4: Sparse face reconstructions under increasing observation rates $\rho (5\%-20\%)$, showing consistent visual improvement, and decreasing MSE.
  • Figure 5: Mean Squared Error (MSE) under increasing Gaussian, Salt-Pepper, and Uniform noise. ULHM achieved lower than the Residual Attention baseline, demonstrating superior robustness.

Theorems & Definitions (9)

  • Definition 1: Bi-Lipschitz Map
  • Theorem 1: Topological Unification
  • Theorem 2: Bi-Lipschitz Sufficiency
  • Proposition 1: Metric-Theoretical Bridge
  • Remark 1: Threshold Selection and Numerical Unity
  • proof
  • proof
  • proof
  • Remark 2