DeepAtlas: a tool for effective manifold learning

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.