Diffusion Earth Mover’s Distance and Distribution Embeddings

Alexander Y Tong, Guillaume Huguet, Amine Natik, Kincaid Macdonald, Manik Kuchroo, Ronald Coifman, Guy Wolf, Smita Krishnaswamy
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:10336-10346, 2021.

Abstract

We propose a new fast method of measuring distances between large numbers of related high dimensional datasets called the Diffusion Earth Mover’s Distance (EMD). We model the datasets as distributions supported on common data graph that is derived from the affinity matrix computed on the combined data. In such cases where the graph is a discretization of an underlying Riemannian closed manifold, we prove that Diffusion EMD is topologically equivalent to the standard EMD with a geodesic ground distance. Diffusion EMD can be computed in {Õ}(n) time and is more accurate than similarly fast algorithms such as tree-based EMDs. We also show Diffusion EMD is fully differentiable, making it amenable to future uses in gradient-descent frameworks such as deep neural networks. Finally, we demonstrate an application of Diffusion EMD to single cell data collected from 210 COVID-19 patient samples at Yale New Haven Hospital. Here, Diffusion EMD can derive distances between patients on the manifold of cells at least two orders of magnitude faster than equally accurate methods. This distance matrix between patients can be embedded into a higher level patient manifold which uncovers structure and heterogeneity in patients. More generally, Diffusion EMD is applicable to all datasets that are massively collected in parallel in many medical and biological systems.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-tong21a, title = {Diffusion Earth Mover’s Distance and Distribution Embeddings}, author = {Tong, Alexander Y and Huguet, Guillaume and Natik, Amine and Macdonald, Kincaid and Kuchroo, Manik and Coifman, Ronald and Wolf, Guy and Krishnaswamy, Smita}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {10336--10346}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/tong21a/tong21a.pdf}, url = {https://proceedings.mlr.press/v139/tong21a.html}, abstract = {We propose a new fast method of measuring distances between large numbers of related high dimensional datasets called the Diffusion Earth Mover’s Distance (EMD). We model the datasets as distributions supported on common data graph that is derived from the affinity matrix computed on the combined data. In such cases where the graph is a discretization of an underlying Riemannian closed manifold, we prove that Diffusion EMD is topologically equivalent to the standard EMD with a geodesic ground distance. Diffusion EMD can be computed in {Õ}(n) time and is more accurate than similarly fast algorithms such as tree-based EMDs. We also show Diffusion EMD is fully differentiable, making it amenable to future uses in gradient-descent frameworks such as deep neural networks. Finally, we demonstrate an application of Diffusion EMD to single cell data collected from 210 COVID-19 patient samples at Yale New Haven Hospital. Here, Diffusion EMD can derive distances between patients on the manifold of cells at least two orders of magnitude faster than equally accurate methods. This distance matrix between patients can be embedded into a higher level patient manifold which uncovers structure and heterogeneity in patients. More generally, Diffusion EMD is applicable to all datasets that are massively collected in parallel in many medical and biological systems.} }
Endnote
%0 Conference Paper %T Diffusion Earth Mover’s Distance and Distribution Embeddings %A Alexander Y Tong %A Guillaume Huguet %A Amine Natik %A Kincaid Macdonald %A Manik Kuchroo %A Ronald Coifman %A Guy Wolf %A Smita Krishnaswamy %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-tong21a %I PMLR %P 10336--10346 %U https://proceedings.mlr.press/v139/tong21a.html %V 139 %X We propose a new fast method of measuring distances between large numbers of related high dimensional datasets called the Diffusion Earth Mover’s Distance (EMD). We model the datasets as distributions supported on common data graph that is derived from the affinity matrix computed on the combined data. In such cases where the graph is a discretization of an underlying Riemannian closed manifold, we prove that Diffusion EMD is topologically equivalent to the standard EMD with a geodesic ground distance. Diffusion EMD can be computed in {Õ}(n) time and is more accurate than similarly fast algorithms such as tree-based EMDs. We also show Diffusion EMD is fully differentiable, making it amenable to future uses in gradient-descent frameworks such as deep neural networks. Finally, we demonstrate an application of Diffusion EMD to single cell data collected from 210 COVID-19 patient samples at Yale New Haven Hospital. Here, Diffusion EMD can derive distances between patients on the manifold of cells at least two orders of magnitude faster than equally accurate methods. This distance matrix between patients can be embedded into a higher level patient manifold which uncovers structure and heterogeneity in patients. More generally, Diffusion EMD is applicable to all datasets that are massively collected in parallel in many medical and biological systems.
APA
Tong, A.Y., Huguet, G., Natik, A., Macdonald, K., Kuchroo, M., Coifman, R., Wolf, G. & Krishnaswamy, S.. (2021). Diffusion Earth Mover’s Distance and Distribution Embeddings. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:10336-10346 Available from https://proceedings.mlr.press/v139/tong21a.html.

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