Heat flow, log-concavity, and Lipschitz transport maps
Brigati G, Pedrotti F. Heat flow, log-concavity, and Lipschitz transport maps. arXiv, 10.48550/arXiv.2404.15205.
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https://doi.org/10.48550/arXiv.2404.15205
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Abstract
In this paper we derive estimates for the Hessian of the logarithm
(log-Hessian) for solutions to the heat equation. For initial data in the form
of log-Lipschitz perturbation of strongly log-concave measures, the log-Hessian
admits an explicit, uniform (in space) lower bound. This yields a new estimate
for the Lipschitz constant of a transport map pushing forward the standard
Gaussian to a measure in this class. Further connections are discussed with
score-based diffusion models and improved Gaussian logarithmic Sobolev
inequalities. Finally, we show that assuming only fast decay of the tails of
the initial datum does not suffice to guarantee uniform log-Hessian upper
bounds.
Publishing Year
Date Published
2024-05-08
Journal Title
arXiv
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Cite this
Brigati G, Pedrotti F. Heat flow, log-concavity, and Lipschitz transport maps. arXiv. doi:10.48550/arXiv.2404.15205
Brigati, G., & Pedrotti, F. (n.d.). Heat flow, log-concavity, and Lipschitz transport maps. arXiv. https://doi.org/10.48550/arXiv.2404.15205
Brigati, Giovanni, and Francesco Pedrotti. “Heat Flow, Log-Concavity, and Lipschitz Transport Maps.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2404.15205.
G. Brigati and F. Pedrotti, “Heat flow, log-concavity, and Lipschitz transport maps,” arXiv. .
Brigati G, Pedrotti F. Heat flow, log-concavity, and Lipschitz transport maps. arXiv, 10.48550/arXiv.2404.15205.
Brigati, Giovanni, and Francesco Pedrotti. “Heat Flow, Log-Concavity, and Lipschitz Transport Maps.” ArXiv, doi:10.48550/arXiv.2404.15205.
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arXiv 2404.15205