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

arXiv: https://arxiv.org/abs/2407.01718 · HTML: https://mlsanity.com/paper/2407.01718/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 103 references that are in the map.
- First posted on arXiv 1 Jul 2024.

## Builds on (most cited first, 30 of 31)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Unsupervised Alignment of Embeddings with Wasserstein Procrustes](https://mlsanity.com/paper/1805.11222/) (2018)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Veridical Data Science](https://mlsanity.com/paper/1901.08152/) (2019)
- [On the Efficiency of Entropic Regularized Algorithms for Optimal Transport](https://mlsanity.com/paper/1906.01437/) (2019)
- [Entropic Optimal Transport: Convergence of Potentials](https://mlsanity.com/paper/2104.11720/) (2021)
- [Spectral convergence of diffusion maps: improved error bounds and an alternative normalisation](https://mlsanity.com/paper/2006.02037/) (2020)
- [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021)
- [Manifold learning with bi-stochastic kernels](https://mlsanity.com/paper/1711.06711/) (2017)
- [Removal of Batch Effects using Distribution-Matching Residual Networks](https://mlsanity.com/paper/1610.04181/) (2016)
- [Convergence of Graph Laplacian with kNN Self-tuned Kernels](https://mlsanity.com/paper/2011.01479/) (2020)
- [Latent common manifold learning with alternating diffusion: analysis and applications](https://mlsanity.com/paper/1602.00078/) (2016)
- [Doubly-Stochastic Normalization of the Gaussian Kernel is Robust to Heteroskedastic Noise](https://mlsanity.com/paper/2006.00402/) (2020)
- [MAGAN: Aligning Biological Manifolds](https://mlsanity.com/paper/1803.00385/) (2018)
- [Kernel Manifold Alignment](https://mlsanity.com/paper/1504.02338/) (2015)
- [SNEkhorn: Dimension Reduction with Symmetric Entropic Affinities](https://mlsanity.com/paper/2305.13797/) (2023)
- [Spectral Analysis Of Weighted Laplacians Arising In Data Clustering](https://mlsanity.com/paper/1909.06389/) (2019)
- [Much Faster Algorithms for Matrix Scaling](https://mlsanity.com/paper/1704.02315/) (2017)
- [Geometric structure of graph Laplacian embeddings](https://mlsanity.com/paper/1901.10651/) (2019)
- [Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach](https://mlsanity.com/paper/2203.00126/) (2022)
- [Diffusion Transport Alignment](https://mlsanity.com/paper/2206.07305/) (2022)
- [Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise](https://mlsanity.com/paper/2206.11386/) (2022)
- [Robust Inference of Manifold Density and Geometry by Doubly Stochastic Scaling](https://mlsanity.com/paper/2209.08004/) (2022)
- [Is your data alignable? Principled and interpretable alignability testing and integration of single-cell data](https://mlsanity.com/paper/2308.01839/) (2023)
- [Harmonic Alignment](https://mlsanity.com/paper/1810.00386/) (2018)
- [Unsupervised Manifold Alignment with Joint Multidimensional Scaling](https://mlsanity.com/paper/2207.02968/) (2022)
- [Manifold Alignment with Label Information](https://mlsanity.com/paper/2210.12774/) (2022)
- [Kernel spectral joint embeddings for high-dimensional noisy datasets using duo-landmark integral operators](https://mlsanity.com/paper/2405.12317/) (2024)
- [Beyond kNN: Adaptive, Sparse Neighborhood Graphs via Optimal Transport](https://mlsanity.com/paper/2208.00604/) (2022)

## Built on by (most cited first, 3 of 3)

- [Stacked SVD or SVD stacked? A Random Matrix Theory perspective on data integration](https://mlsanity.com/paper/2507.22170/) (2025)
- [Enhancing Spectral Embedding through Robust and Flexible Knowledge Transfer in Electronic Health Records](https://mlsanity.com/paper/2606.11570/) (2026)
- [Density-Reweighted Entropic Optimal Transport: Decoupling Geometry from Sampling Density](https://mlsanity.com/paper/2608.16506/) (2026)

## Abstract (arXiv)

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 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 them 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 analyze a generative model in which two observed high-dimensional datasets share latent variables supported on a common low-dimensional manifold, while each dataset is subject to translation, geometric distortion, orthogonal nuisance structure, and noise. In a large-sample, high-dimensional regime, we prove that the EOT plan concentrates around a population kernel on an effective manifold determined by the geometric mean of the distortions, with invariance to translations, orthogonal nuisance structure, and noise. Subsequently, we relate our embedding to 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.