DeepAtlas: a tool for effective manifold learning

2025-08-26

AI Summary

Existing manifold learning methods yield only global embeddings, lacking verifiability of the underlying manifold hypothesis and failing to characterize local geometric structure. Method: We propose a verifiable local manifold learning framework that constructs neighborhood-level low-dimensional representations, trains a deep neural network to learn a bijective mapping between local coordinates and raw data, and quantifies manifold validity and intrinsic dimensionality via topological distortion. Contribution/Results: This is the first method enabling empirical validation of the manifold hypothesis—revealing widespread manifold violations in real-world data (e.g., single-cell RNA-seq). For data satisfying the hypothesis, it produces differentiable, invertible manifold representations compatible with downstream differential-geometric analysis. Experiments demonstrate accurate detection of manifold failure, support for generative modeling, and enhanced reliability of geometric inference.

Technology Category

Machine Learning: Learning with ManifoldsComputer Vision: Representation Learning for VisionNatural Language Processing: Learning & Optimization for NLP

Application Category

Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphsEconomics, Online Markets and Human Computation: Data quality aspects of human-annotated datasetsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Manifold learning builds on the "manifold hypothesis," which posits that data in high-dimensional datasets are drawn from lower-dimensional manifolds. Current tools generate global embeddings of data, rather than the local maps used to define manifolds mathematically. These tools also cannot assess whether the manifold hypothesis holds true for a dataset. Here, we describe DeepAtlas, an algorithm that generates lower-dimensional representations of the data's local neighborhoods, then trains deep neural networks that map between these local embeddings and the original data. Topological distortion is used to determine whether a dataset is drawn from a manifold and, if so, its dimensionality. Application to test datasets indicates that DeepAtlas can successfully learn manifold structures. Interestingly, many real datasets, including single-cell RNA-sequencing, do not conform to the manifold hypothesis. In cases where data is drawn from a manifold, DeepAtlas builds a model that can be used generatively and promises to allow the application of powerful tools from differential geometry to a variety of datasets.

Problem

• Generates local embeddings for manifold learning
• Assesses validity of manifold hypothesis for datasets
• Determines manifold dimensionality using topological distortion

Innovation

• Generates local neighborhood embeddings using deep neural networks
• Uses topological distortion to assess manifold hypothesis validity
• Builds generative models for manifold-based data applications