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Sparsity of Quadratically Regularized Optimal Transport: Bounds on concentration and bias

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arxiv 2410.03425 v1 pith:GOBJ5ULM submitted 2024-10-04 math.OC math.PR

Johannes Wiesel , Xingyu Xu This is my paper · ORCID
classification math.OCmath.PR
keywords epsilonoptimalboundstransportcouplingproblemquadraticquadratically
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abstract

We study the quadratically regularized optimal transport (QOT) problem for quadratic cost and compactly supported marginals $\mu$ and $\nu$. It has been empirically observed that the optimal coupling $\pi_\epsilon$ for the QOT problem has sparse support for small regularization parameter $\epsilon>0.$ In this article we provide the first quantitative description of this phenomenon in general dimension: we derive bounds on the size and on the location of the support of $\pi_\epsilon$ compared to the Monge coupling. Our analysis is based on pointwise bounds on the density of $\pi_\epsilon$ together with Minty's trick, which provides a quadratic detachment from the optimal transport duality gap. In the self-transport setting $\mu=\nu$ we obtain optimal rates of order $\epsilon^{\frac{1}{2+d}}.$

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Sparse Regularized Optimal Transport without Curse of Dimensionality

    math.ST 2025-05 accept novelty 8.0 of 10

    Sparse f-divergence regularized optimal transport estimators converge at parametric rate with central limit theorems, independent of dimension.

  2. Quadratic-form Optimal Transport

    math.PR 2025-01 accept novelty 8.0 of 10

    Quadratic-form optimal transport is introduced, and for several cost classes including the rectangular cost, the unique minimizer is a new diamond-shaped coupling rather than the usual comonotone or antimonotone couplings.

  3. Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives

    math.ST 2025-06 conditional

    A survey of minimax rates and limit laws for estimating optimal transport maps from samples, covering smooth, Gaussian, semi-discrete, entropic, and divergence-regularized settings.

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