Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps

Marco Cuturi, Michal Klein, Pierre Ablin
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:6671-6682, 2023.

Abstract

Optimal transport (OT) theory focuses, among all maps $T:\mathbb{R}^d\rightarrow \mathbb{R}^d$ that can morph a probability measure $\mu$ onto another $\nu$, on those that are the “thriftiest”, i.e. such that the average cost $c(x, T(x))$ between $x$ and its image $T(x)$ is as small as possible. Many computational approaches have been proposed to estimate such Monge maps when $c$ is the squared-Euclidean distance, e.g., using entropic maps [Pooladian+2021], or input convex neural networks [Makkuva+2020, Korotin+2020]. We propose a new research direction, that leverages a specific translation invariant cost $c(x, y):=h(x-y)$ inspired by the elastic net. Here, $h:=\tfrac{1}{2}\|\cdot\|_2^2+\tau(\cdot)$, where $\tau$ is a convex function. We highlight a surprising link tying together a generalized entropic map for $h$, Bregman centroids induced by $h$, and the proximal operator of $\tau$. We show how setting $\tau$ to be a sparsity-inducing norm results in the first application of Occam’s razor to transport. These maps yield, mechanically, displacement vectors $\Delta(x):= T(x)-x$ that are sparse, with sparsity patterns that vary depending on $x$. We showcase the ability of our method to estimate meaningful OT maps for high-dimensional single-cell transcription data. We use our methods in the $34000$-d space of gene counts for cells, without using a prior dimensionality reduction, thus retaining the ability to interpret all displacements at the gene level.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-cuturi23a, title = {Monge, {B}regman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps}, author = {Cuturi, Marco and Klein, Michal and Ablin, Pierre}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {6671--6682}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/cuturi23a/cuturi23a.pdf}, url = {https://proceedings.mlr.press/v202/cuturi23a.html}, abstract = {Optimal transport (OT) theory focuses, among all maps $T:\mathbb{R}^d\rightarrow \mathbb{R}^d$ that can morph a probability measure $\mu$ onto another $\nu$, on those that are the “thriftiest”, i.e. such that the average cost $c(x, T(x))$ between $x$ and its image $T(x)$ is as small as possible. Many computational approaches have been proposed to estimate such Monge maps when $c$ is the squared-Euclidean distance, e.g., using entropic maps [Pooladian+2021], or input convex neural networks [Makkuva+2020, Korotin+2020]. We propose a new research direction, that leverages a specific translation invariant cost $c(x, y):=h(x-y)$ inspired by the elastic net. Here, $h:=\tfrac{1}{2}\|\cdot\|_2^2+\tau(\cdot)$, where $\tau$ is a convex function. We highlight a surprising link tying together a generalized entropic map for $h$, Bregman centroids induced by $h$, and the proximal operator of $\tau$. We show how setting $\tau$ to be a sparsity-inducing norm results in the first application of Occam’s razor to transport. These maps yield, mechanically, displacement vectors $\Delta(x):= T(x)-x$ that are sparse, with sparsity patterns that vary depending on $x$. We showcase the ability of our method to estimate meaningful OT maps for high-dimensional single-cell transcription data. We use our methods in the $34000$-d space of gene counts for cells, without using a prior dimensionality reduction, thus retaining the ability to interpret all displacements at the gene level.} }
Endnote
%0 Conference Paper %T Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps %A Marco Cuturi %A Michal Klein %A Pierre Ablin %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-cuturi23a %I PMLR %P 6671--6682 %U https://proceedings.mlr.press/v202/cuturi23a.html %V 202 %X Optimal transport (OT) theory focuses, among all maps $T:\mathbb{R}^d\rightarrow \mathbb{R}^d$ that can morph a probability measure $\mu$ onto another $\nu$, on those that are the “thriftiest”, i.e. such that the average cost $c(x, T(x))$ between $x$ and its image $T(x)$ is as small as possible. Many computational approaches have been proposed to estimate such Monge maps when $c$ is the squared-Euclidean distance, e.g., using entropic maps [Pooladian+2021], or input convex neural networks [Makkuva+2020, Korotin+2020]. We propose a new research direction, that leverages a specific translation invariant cost $c(x, y):=h(x-y)$ inspired by the elastic net. Here, $h:=\tfrac{1}{2}\|\cdot\|_2^2+\tau(\cdot)$, where $\tau$ is a convex function. We highlight a surprising link tying together a generalized entropic map for $h$, Bregman centroids induced by $h$, and the proximal operator of $\tau$. We show how setting $\tau$ to be a sparsity-inducing norm results in the first application of Occam’s razor to transport. These maps yield, mechanically, displacement vectors $\Delta(x):= T(x)-x$ that are sparse, with sparsity patterns that vary depending on $x$. We showcase the ability of our method to estimate meaningful OT maps for high-dimensional single-cell transcription data. We use our methods in the $34000$-d space of gene counts for cells, without using a prior dimensionality reduction, thus retaining the ability to interpret all displacements at the gene level.
APA
Cuturi, M., Klein, M. & Ablin, P.. (2023). Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:6671-6682 Available from https://proceedings.mlr.press/v202/cuturi23a.html.

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