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

• URL: http://arxiv.org/abs/2601.09025v1
• Date: Tue, 13 Jan 2026 23:08:16 GMT
• Title: Universal Latent Homeomorphic Manifolds: Cross-Domain Representation Learning via Homeomorphism Verification
• Authors: Tong Wu, Tayab Uddin Wara, Daniel Hernandez, Sidong Lei,
• Abstract summary: We present a framework that unifies semantic representations and observation-driven machine representations into a single latent structure.<n>We use emphhomeomorphism as a criterion for determining when latent manifold induced by different semantic-observation pairs can be rigorously unified.<n>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.
• Score: 4.509161738293017
• 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.

Related papers

• Distributional Equivalence in Linear Non-Gaussian Latent-Variable Cyclic Causal Models: Characterization and Learning [13.891913455492697]
  We argue that a core obstacle to a general, structural-assumption-free approach is the lack of an equivalence characterization.<n>Key to our approach is a new tool, edge rank constraints, which fills a missing piece in the toolbox for latent-variable causal discovery.
  arXiv  Detail & Related papers  (2026-03-05T03:57:14Z)
• Scale-Consistent State-Space Dynamics via Fractal of Stationary Transformations [9.983526161001997]
  Recent deep learning models increasingly rely on depth without structural guarantees on the validity of intermediate representations.<n>We address this limitation by formulating a structural requirement for state-space model's scale-consistent latent dynamics.<n>We empirically verify the predicted scale-consistent behavior, showing that adaptive efficiency emerges from the aligned latent geometry.
  arXiv  Detail & Related papers  (2026-01-27T12:44:20Z)
• CoPHo: Classifier-guided Conditional Topology Generation with Persistent Homology [14.522233245543687]
  Topology structure underpins research on performance and robustness.<n>Generation of synthetic graphs with desired properties for testing or release.<n>We propose Persistent Topology Generation with Conditional Homology (CoPHo)<n>Experiments on four generic/network datasets demonstrate that CoPHo outperforms existing methods at matching target metrics.
  arXiv  Detail & Related papers  (2025-12-17T13:10:22Z)
• Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction [54.95522167029998]
  This article is a self-contained primer on diffusion over general state spaces.<n>We develop the discrete-time view (forward noising via Markov kernels and learned reverse dynamics) alongside its continuous-time limits.<n>A common variational treatment yields the ELBO that underpins standard training losses.
  arXiv  Detail & Related papers  (2025-12-04T18:55:36Z)
• Aligned Manifold Property and Topology Point Clouds for Learning Molecular Properties [55.2480439325792]
  This work introduces AMPTCR, a molecular surface representation that combines local quantum-derived scalar fields and custom topological descriptors within an aligned point cloud format.<n>For molecular weight, results confirm that AMPTCR encodes physically meaningful data, with a validation R2 of 0.87.<n>In the bacterial inhibition task, AMPTCR enables both classification and direct regression of E. coli inhibition values.
  arXiv  Detail & Related papers  (2025-07-22T04:35:50Z)
• Structure-Aware Correspondence Learning for Relative Pose Estimation [65.44234975976451]
  Relative pose estimation provides a promising way for achieving object-agnostic pose estimation.<n>Existing 3D correspondence-based methods suffer from small overlaps in visible regions and unreliable feature estimation for invisible regions.<n>We propose a novel Structure-Aware Correspondence Learning method for Relative Pose Estimation, which consists of two key modules.
  arXiv  Detail & Related papers  (2025-03-24T13:43:44Z)
• STITCH: Surface reconstrucTion using Implicit neural representations with Topology Constraints and persistent Homology [23.70495314317551]
  We present STITCH, a novel approach for neural implicit surface reconstruction of a sparse and irregularly spaced point cloud.<n>We develop a new differentiable framework based on persistent homology to formulate topological loss terms that enforce the prior of a single 2-manifold object.
  arXiv  Detail & Related papers  (2024-12-24T22:55:35Z)
• PseudoNeg-MAE: Self-Supervised Point Cloud Learning using Conditional Pseudo-Negative Embeddings [55.55445978692678]
  PseudoNeg-MAE enhances global feature representation of point cloud masked autoencoders by making them both discriminative and sensitive to transformations.<n>We propose a novel loss that explicitly penalizes invariant collapse, enabling the network to capture richer transformation cues while preserving discriminative representations.
  arXiv  Detail & Related papers  (2024-09-24T07:57:21Z)
• Relative Representations: Topological and Geometric Perspectives [50.85040046976025]
  Relative representations are an established approach to zero-shot model stitching.<n>We introduce a normalization procedure in the relative transformation, resulting in invariance to non-isotropic rescalings and permutations.<n>Second, we propose to deploy topological densification when fine-tuning relative representations, a topological regularization loss encouraging clustering within classes.
  arXiv  Detail & Related papers  (2024-09-17T08:09:22Z)
• From Bricks to Bridges: Product of Invariances to Enhance Latent Space Communication [19.336940758147442]
  It has been observed that representations learned by distinct neural networks conceal structural similarities when the models are trained under similar inductive biases. We introduce a versatile method to directly incorporate a set of invariances into the representations, constructing a product space of invariant components on top of the latent representations. We validate our solution on classification and reconstruction tasks, observing consistent latent similarity and downstream performance improvements in a zero-shot stitching setting.
  arXiv  Detail & Related papers  (2023-10-02T13:55:38Z)
• Towards Characterizing Domain Counterfactuals For Invertible Latent Causal Models [15.817239008727789]
  In this work, we analyze a specific type of causal query called domain counterfactuals, which hypothesizes what a sample would have looked like if it had been generated in a different domain. We show that recovering the latent Structural Causal Model (SCM) is unnecessary for estimating domain counterfactuals. We also develop a theoretically grounded practical algorithm that simplifies the modeling process to generative model estimation.
  arXiv  Detail & Related papers  (2023-06-20T04:19:06Z)
• GSMFlow: Generation Shifts Mitigating Flow for Generalized Zero-Shot Learning [55.79997930181418]
  Generalized Zero-Shot Learning aims to recognize images from both the seen and unseen classes by transferring semantic knowledge from seen to unseen classes. It is a promising solution to take the advantage of generative models to hallucinate realistic unseen samples based on the knowledge learned from the seen classes. We propose a novel flow-based generative framework that consists of multiple conditional affine coupling layers for learning unseen data generation.
  arXiv  Detail & Related papers  (2022-07-05T04:04:37Z)
• Surreal-GAN:Semi-Supervised Representation Learning via GAN for uncovering heterogeneous disease-related imaging patterns [4.965264481651854]
  We propose Surreal-GAN (Semi-SUpeRvised ReprEsentAtion Learning via GAN) to model disease effects. We validated the model through extensive semi-synthetic experiments, and then demonstrate its potential in capturing biologically plausible imaging patterns in Alzheimer's disease.
  arXiv  Detail & Related papers  (2022-05-09T19:09:28Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.