Structured Transforms Across Spaces with Cost-Regularized Optimal Transport

Othmane Sebbouh, Marco Cuturi, Gabriel Peyré
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:586-594, 2024.

Abstract

Matching a source to a target probability measure is often solved by instantiating a linear optimal transport (OT) problem, parameterized by a ground cost function that quantifies discrepancy between points. When these measures live in the same metric space, the ground cost often defaults to its distance. When instantiated across two different spaces, however, choosing that cost in the absence of aligned data is a conundrum. As a result, practitioners often resort to solving instead a quadratic Gromow-Wasserstein (GW) problem. We exploit in this work a parallel between GW and cost-regularized OT, the regularized minimization of a linear OT objective parameterized by a ground cost. We use this cost-regularized formulation to match measures across two different Euclidean spaces, where the cost is evaluated between transformed source points and target points. We show that several quadratic OT problems fall in this category, and consider enforcing structure in linear transform (e.g., sparsity), by introducing structure-inducing regularizers. We provide a proximal algorithm to extract such transforms from unaligned data, and demonstrate its applicability to single-cell spatial transcriptomics/multiomics matching tasks.

Cite this Paper


BibTeX
@InProceedings{pmlr-v238-sebbouh24a, title = {Structured Transforms Across Spaces with Cost-Regularized Optimal Transport}, author = {Sebbouh, Othmane and Cuturi, Marco and Peyr\'{e}, Gabriel}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {586--594}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/sebbouh24a/sebbouh24a.pdf}, url = {https://proceedings.mlr.press/v238/sebbouh24a.html}, abstract = {Matching a source to a target probability measure is often solved by instantiating a linear optimal transport (OT) problem, parameterized by a ground cost function that quantifies discrepancy between points. When these measures live in the same metric space, the ground cost often defaults to its distance. When instantiated across two different spaces, however, choosing that cost in the absence of aligned data is a conundrum. As a result, practitioners often resort to solving instead a quadratic Gromow-Wasserstein (GW) problem. We exploit in this work a parallel between GW and cost-regularized OT, the regularized minimization of a linear OT objective parameterized by a ground cost. We use this cost-regularized formulation to match measures across two different Euclidean spaces, where the cost is evaluated between transformed source points and target points. We show that several quadratic OT problems fall in this category, and consider enforcing structure in linear transform (e.g., sparsity), by introducing structure-inducing regularizers. We provide a proximal algorithm to extract such transforms from unaligned data, and demonstrate its applicability to single-cell spatial transcriptomics/multiomics matching tasks.} }
Endnote
%0 Conference Paper %T Structured Transforms Across Spaces with Cost-Regularized Optimal Transport %A Othmane Sebbouh %A Marco Cuturi %A Gabriel Peyré %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-sebbouh24a %I PMLR %P 586--594 %U https://proceedings.mlr.press/v238/sebbouh24a.html %V 238 %X Matching a source to a target probability measure is often solved by instantiating a linear optimal transport (OT) problem, parameterized by a ground cost function that quantifies discrepancy between points. When these measures live in the same metric space, the ground cost often defaults to its distance. When instantiated across two different spaces, however, choosing that cost in the absence of aligned data is a conundrum. As a result, practitioners often resort to solving instead a quadratic Gromow-Wasserstein (GW) problem. We exploit in this work a parallel between GW and cost-regularized OT, the regularized minimization of a linear OT objective parameterized by a ground cost. We use this cost-regularized formulation to match measures across two different Euclidean spaces, where the cost is evaluated between transformed source points and target points. We show that several quadratic OT problems fall in this category, and consider enforcing structure in linear transform (e.g., sparsity), by introducing structure-inducing regularizers. We provide a proximal algorithm to extract such transforms from unaligned data, and demonstrate its applicability to single-cell spatial transcriptomics/multiomics matching tasks.
APA
Sebbouh, O., Cuturi, M. & Peyré, G.. (2024). Structured Transforms Across Spaces with Cost-Regularized Optimal Transport. Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:586-594 Available from https://proceedings.mlr.press/v238/sebbouh24a.html.

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