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On Optimal Transport Maps Between 1 /d-Concave Densities
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abstract
In this paper, we extend the scope of Caffarelli's contraction theorem, which provides a measure of the Lipschitz constant for optimal transport maps between log-concave probability densities in $\R^d$. Our focus is on a broader category of densities, specifically those that are $\nicefrac{1}{d}$-concave and can be represented as $V^{-d}$, where $V$ is convex. By setting appropriate conditions, we derive linear or sublinear limitations for the optimal transport map. This leads us to a comprehensive Lipschitz estimate that aligns with the principles established in Caffarelli's theorem.
Forward citations
Cited by 3 Pith papers
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Spectral Obstructions to Contracting Transport Maps on Curved Spaces
Explicit metrics on S^d and weighted R^d for d≥4 violate Milman's spectral comparison conjectures, ruling out 1-Lipschitz transport maps; the abstract's extra dimension-2 and hemisphere results are not in the text.
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Exponential convergence for ultrafast diffusion equations with log-concave weights
For weighted ultrafast diffusion equations on R with log-concave, log-Lipschitz weights, solutions starting close to equilibrium converge exponentially fast in relative L2.
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Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives
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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