# Latent Manifold Reconstruction and Representation with Topological and Geometrical Regularization

arXiv: https://arxiv.org/abs/2505.04412 · HTML: https://mlsanity.com/paper/2505.04412/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 37 references that are in the map.
- First posted on arXiv 7 May 2025.

## Builds on (most cited first, 14 of 14)

- [PartNet: A Large-scale Benchmark for Fine-grained and Hierarchical Part-level 3D Object Understanding](https://mlsanity.com/paper/1812.02713/) (2018)
- [Tune: A Research Platform for Distributed Model Selection and Training](https://mlsanity.com/paper/1807.05118/) (2018)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Parametric UMAP embeddings for representation and semi-supervised learning](https://mlsanity.com/paper/2009.12981/) (2020)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Representation Topology Divergence: A Method for Comparing Neural Network Representations](https://mlsanity.com/paper/2201.00058/) (2021)
- [Learning Topology-Preserving Data Representations](https://mlsanity.com/paper/2302.00136/) (2023)
- [Geometric Autoencoders -- What You See is What You Decode](https://mlsanity.com/paper/2306.17638/) (2023)
- [Isometric Representation Learning for Disentangled Latent Space of Diffusion Models](https://mlsanity.com/paper/2407.11451/) (2024)
- [Differentiable Euler Characteristic Transforms for Shape Classification](https://mlsanity.com/paper/2310.07630/) (2023)
- [Representation Learning via Manifold Flattening and Reconstruction](https://mlsanity.com/paper/2305.01777/) (2023)
- [Asymptotic theory for density ridges](https://mlsanity.com/paper/1406.5663/) (2014)
- [An Invitation to the Euler Characteristic Transform](https://mlsanity.com/paper/2310.10395/) (2023)

## Built on by (most cited first, 1 of 1)

- [Universal Latent Homeomorphic Manifolds: A Framework for Cross-Domain Representation Unification](https://mlsanity.com/paper/2601.09025/) (2026)

## Abstract (arXiv)

Manifold learning aims to discover and represent low-dimensional structures underlying high-dimensional data while preserving critical topological and geometric properties. Existing methods often fail to capture local details with global topological integrity from noisy data or construct a balanced dimensionality reduction, resulting in distorted or fractured embeddings. We present an AutoEncoder-based method that integrates a manifold reconstruction layer, which uncovers latent manifold structures from noisy point clouds, and further provides regularizations on topological and geometric properties during dimensionality reduction, whereas the two components promote each other during training. Experiments on point cloud datasets demonstrate that our method outperforms baselines like t-SNE, UMAP, and Topological AutoEncoders in discovering manifold structures from noisy data and preserving them through dimensionality reduction, as validated by visualization and quantitative metrics. This work demonstrates the significance of combining manifold reconstruction with manifold learning to achieve reliable representation of the latent manifold, particularly when dealing with noisy real-world data. Code repository: https://github.com/Thanatorika/mrtg.