DeepAtlas: a tool for effective manifold learning

Area of Science:

• Computational topology
• Machine learning
• Data science

Background:

• Manifold learning assumes high-dimensional data resides on lower-dimensional manifolds.
• Existing methods produce global embeddings, not local maps needed for manifold definition.
• Current tools cannot validate the manifold hypothesis for a given dataset.

Purpose of the Study:

• Introduce DeepAtlas, an algorithm for learning local data structures.
• Enable assessment of the manifold hypothesis's validity.
• Facilitate generative modeling and differential geometry on manifold data.

Main Methods:

• Generate low-dimensional local neighborhood embeddings.
• Train deep neural networks for mapping between local embeddings and original data.
• Utilize topological distortion to assess manifold properties and dimensionality.

Main Results:

• DeepAtlas successfully learns manifold structures in test datasets.
• Demonstrated that many real-world datasets, including single-cell RNA-sequencing, do not adhere to the manifold hypothesis.
• Developed a generative model for datasets conforming to the manifold hypothesis.

Conclusions:

• DeepAtlas provides a novel approach to manifold learning and hypothesis validation.
• Highlights the limitations of the manifold hypothesis for certain complex datasets.
• Opens avenues for applying differential geometry to diverse data types.