Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasets

TL;DR

This paper introduces Entropic Optimal Transport eigenmaps, a novel method for aligning and jointly embedding high-dimensional datasets with shared underlying structures, providing theoretical guarantees and demonstrating superior performance in real-world biological data analysis.

Contribution

The paper proposes a new EOT eigenmap technique that aligns and embeds datasets by leveraging singular vectors of the EOT plan, with theoretical analysis and practical validation.

Findings

EOT eigenmaps effectively recover shared manifold structures.

The method outperforms existing techniques in challenging biological data scenarios.

The approach has solid theoretical guarantees in high-dimensional regimes.

Abstract

Embedding high-dimensional data into a low-dimensional space is an indispensable component of data analysis. In numerous applications, it is necessary to align and jointly embed multiple datasets from different studies or experimental conditions. Such datasets may share underlying structures of interest but exhibit individual distortions, resulting in misaligned embeddings using traditional techniques. In this work, we propose \textit{Entropic Optimal Transport (EOT) eigenmaps}, a principled approach for aligning and jointly embedding a pair of datasets with theoretical guarantees. Our approach leverages the leading singular vectors of the EOT plan matrix between two datasets to extract their shared underlying structure and align the datasets accordingly in a common embedding space. We interpret our approach as an inter-data variant of the classical Laplacian eigenmaps and diffusion maps embeddings, showing that it enjoys many favorable analogous properties. We then analyze a data-generative model where two observed high-dimensional datasets share latent variables on a common low-dimensional manifold, but each dataset is subject to data-specific translation, scaling, nuisance structures, and noise. We show that in a high-dimensional asymptotic regime, the EOT plan recovers the shared manifold structure by approximating a kernel function evaluated at the locations of the latent variables. Subsequently, we provide a geometric interpretation of our embedding by relating it to the eigenfunctions of population-level operators encoding the density and geometry of the shared manifold. Finally, we showcase the performance of our approach for data integration and embedding through simulations and analyses of real-world biological data, demonstrating its advantages over alternative methods in challenging scenarios.