Latent Manifold Reconstruction and Representation with Topological and Geometrical Regularization

TL;DR

The paper addresses the challenge of preserving both local geometry and global topology when embedding noisy high-dimensional data by introducing an AutoEncoder augmented with a Manifold Reconstruction Layer (MRL) and two regularizers. The topological regularizer leverages persistent homology via Vietoris-Rips filtrations to align persistent diagrams between the latent manifold and the embedding, while the geometric regularizer enforces a near-isometric mapping through a coordinate-invariant functional based on the pullback metric. Training is end-to-end, with a total loss of $L = \lambda_{AE}L_{AE} + \lambda_{topo}L_{topo} + \lambda_{geom}L_{geom}$, enabling mutual reinforcement between manifold reconstruction and representation learning. Experiments on Swiss roll, Mammoth, PartNet, and high-dimensional spheres show that the proposed method maintains both local and global structures more accurately than baselines like t-SNE, UMAP, and TopoAE, across visualizations and quantitative metrics. The work advances robust latent-manifold representation in noisy data and provides a practical codebase for researchers and practitioners working on manifold learning and visualizations in real-world settings.

Abstract

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.

Figures (11)

• Figure 1: (a) An illustration of our model pipeline, (b) an illustration of our manifold reconstruction algorithm, (c) an illustration of the topological regularizer based on persistent homology, (d) an illustration of the geometric regularizer based on scaled isometry.
• Figure 2: Dimensionality reduction of 3D point cloud data to 2D space. Our proposed method best preserves global and local structures of the latent manifolds underlying noisy data.
• Figure 3: Dimensionality reduction of 100-D spheres, and "rocket" object from PartNet dataset.
• Figure S1: Dimensionality reduction results of a 12-object subset of the PartNet dataset. The rows are the original 3D point cloud of the object, and 5 different manifold learning methods to be compared. The columns are 12 different objects, with ground-truth semantic segmentation of different parts.
• Figure S2: The visualization of our ablation study. (a) is the implicit ground-truth Swiss roll point cloud; (b) is the noisy 3D point cloud, which is the input data of our models; (c) and (d) are reconstructed latent manifolds with our Final model and MR AE, respectively; (e)-(j) are the embeddings of various models to compare in the ablation study.

Theorems & Definitions (22)

• Lemma 1
• Definition 1
• Definition 2
• Definition 3
• Definition C1
• Definition C2
• Definition C3
• Definition C4
• Theorem C1
• Definition C5