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

arXiv: https://arxiv.org/abs/2601.09025 · HTML: https://mlsanity.com/paper/2601.09025/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 52 of its 68 references that are in the map.
- First posted on arXiv 13 Jan 2026.

## Builds on (most cited first, 30 of 52)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
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- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Prototypical Networks for Few-shot Learning](https://mlsanity.com/paper/1703.05175/) (2017)
- [CBAM: Convolutional Block Attention Module](https://mlsanity.com/paper/1807.06521/) (2018)
- [SwinIR: Image Restoration Using Swin Transformer](https://mlsanity.com/paper/2108.10257/) (2021)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Unsupervised Representation Learning by Predicting Image Rotations](https://mlsanity.com/paper/1803.07728/) (2018)
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- [Deep CORAL: Correlation Alignment for Deep Domain Adaptation](https://mlsanity.com/paper/1607.01719/) (2016)
- [Editing Models with Task Arithmetic](https://mlsanity.com/paper/2212.04089/) (2022)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Dual Attention Network for Scene Segmentation](https://mlsanity.com/paper/1809.02983/) (2018)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [TIES-Merging: Resolving Interference When Merging Models](https://mlsanity.com/paper/2306.01708/) (2023)
- [Maximum Classifier Discrepancy for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1712.02560/) (2017)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Image Inpainting for Irregular Holes Using Partial Convolutions](https://mlsanity.com/paper/1804.07723/) (2018)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Deep metric learning using Triplet network](https://mlsanity.com/paper/1412.6622/) (2014)
- [Residual Attention Network for Image Classification](https://mlsanity.com/paper/1704.06904/) (2017)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)

## Abstract (arXiv)

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 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.