Entropic Optimal Transport Eigenmaps for Nonlinear Alignment and Joint Embedding of High-Dimensional Datasets
Boris Landa, Yuval Kluger, Rong Ma
TL;DR
This work develops Entropic Optimal Transport (EOT) eigenmaps to align and jointly embed two high-dimensional datasets that share underlying structure but exhibit dataset-specific distortions. By computing the EOT plan W between the datasets and extracting the leading singular vectors, the method yields a common embedding that preserves shared geometry while filtering batch effects; it naturally recovers classical Laplacian eigenmaps when t = 0 and diffusion maps when t is a positive integer. The authors provide theoretical guarantees under a latent manifold model with distortions, showing concentration of W to a population kernel and convergence of the associated operators to a density-weighted Laplacian, enabling robust extraction of the shared manifold structure in high dimensions. Empirically, EOT eigenmaps outperform existing methods in simulated alignment and clustering tasks and demonstrate strong performance in real single-cell data integrations, including multi-omics and cross-modality analyses, with publicly available R/Python implementations. These results have practical impact for data integration and multi-view analyses in biology and beyond, offering a principled, interpretable framework with theoretical support for challenging cross-study distortions.
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.