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@article{mielkeNotesHellingerDistance2025,
title={Some notes on the {Hellinger} distance and various {Fisher-Rao} distances},
author={Mielke, Alexander},
journal={arXiv preprint arXiv:2510.02537},
year={2025}
}
@article{carrillo2024fisher,
title = {{Fisher-Rao} gradient flow: geodesic convexity and functional inequalities},
author = {Carrillo, Jos{\'e} A and Chen, Yifan and Huang, Daniel Zhengyu and Huang, Jiaoyang and Wei, Dongyi},
journal = {arXiv preprint arXiv:2407.15693},
year = {2024}
}
@article{chen2023sampling,
title = {Sampling via gradient flows in the space of probability measures},
author = {Chen, Yifan and Huang, Daniel Zhengyu and Huang, Jiaoyang and Reich, Sebastian and Stuart, Andrew M},
journal = {arXiv preprint arXiv:2310.03597},
year = {2023}
}
@book{ambrosio2008gradient,
title = {Gradient flows: In metric spaces and in the space of probability measures},
author = {Ambrosio, Luigi and Gigli, Nicola and Savare, Giuseppe},
year = {2005},
publisher = {Springer Science \& Business Media}
}
@article{delageDistributionallyRobustOptimization2010,
title = {Distributionally robust optimization under moment uncertainty with application to data-driven problems},
author = {Delage, Erick and Ye, Yinyu},
year = {2010},
journal = {Operations research},
volume = {58},
number = {3},
pages = {595--612},
publisher = {INFORMS}
}
@article{ben-talRobustSolutionsOptimization2013,
title = {Robust solutions of optimization problems affected by uncertain probabilities},
author = {{Ben-Tal}, Aharon and {den Hertog}, Dick and De Waegenaere, Anja and Melenberg, Bertrand and Rennen, Gijs},
year = {2013},
month = feb,
journal = {Management Science},
volume = {59},
number = {2},
pages = {341--357},
issn = {0025-1909, 1526-5501},
doi = {10.1287/mnsc.1120.1641},
urldate = {2022-03-02},
langid = {english}
}
@article{levyLargeScaleMethodsDistributionally2020,
title={Large-scale methods for distributionally robust optimization},
author={Levy, Daniel and Carmon, Yair and Duchi, John C and Sidford, Aaron},
journal={Advances in neural information processing systems},
volume={33},
pages={8847--8860},
year={2020}
}
@article{kuhnDistributionallyRobustOptimization2024,
title={Distributionally robust optimization},
author={Kuhn, Daniel and Shafiee, Soroosh and Wiesemann, Wolfram},
journal={Acta Numerica},
volume={34},
pages={579--804},
year={2025},
publisher={Cambridge University Press}
}
@article{gallouetJKOSplittingScheme2018,
title = {A {JKO} splitting scheme for {Kantorovich-Fisher-Rao} gradient flows},
author = {Gallou{\"e}t, Thomas and Monsaingeon, L{\'e}onard},
year = {2018},
month = may,
number = {arXiv:1602.04457},
eprint = {1602.04457},
primaryclass = {math},
publisher = {arXiv},
doi = {10.48550/arXiv.1602.04457},
urldate = {2023-01-15},
archiveprefix = {arXiv},
keywords = {35K15 35K57 35K65 47J30,Mathematics - Analysis of PDEs}
}
@inproceedings{yuFastDistributionallyRobust2022,
title = {Fast distributionally robust learning with variance-reduced min-max optimization},
booktitle = {Proceedings of {{The}} 25th {{International Conference}} on {{Artificial Intelligence}} and {{Statistics}}},
author = {Yu, Yaodong and Lin, Tianyi and Mazumdar, Eric V. and Jordan, Michael},
year = {2022},
month = may,
pages = {1219--1250},
publisher = {PMLR},
issn = {2640-3498},
urldate = {2022-07-22},
langid = {english}
}
@inproceedings{congerStrategicDistributionShift2023,
title = {Strategic distribution shift of interacting agents via coupled gradient flows},
booktitle = {Thirty-Seventh {{Conference}} on {{Neural Information Processing Systems}}},
author = {Conger, Lauren E. and Hoffman, Franca and Mazumdar, Eric and Ratliff, Lillian J.},
year = {2023},
month = nov,
urldate = {2023-12-21},
langid = {english}
}
@article{trillosAdversarialRobustnessUse2023,
title={On adversarial robustness and the use of {Wasserstein} ascent-descent dynamics to enforce it},
author={Garc{\'\i}a Trillos, Camilo Andr{\'e}s and Garc{\'\i}a Trillos, Nicol{\'a}s},
journal={Information and Inference: A Journal of the IMA},
volume={13},
number={3},
pages={iaae018},
year={2024},
publisher={Oxford University Press}
}
@article{luAcceleratingLangevinSampling2019,
title={Accelerating {Langevin} sampling with birth-death},
author={Lu, Yulong and Lu, Jianfeng and Nolen, James},
journal={arXiv preprint arXiv:1905.09863},
year={2019}
}
@article{luBirthdeathDynamicsSampling2023,
title = {Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics},
shorttitle = {Birth-Death Dynamics for Sampling},
author = {Lu, Yulong and Slep{\v c}ev, Dejan and Wang, Lihan},
year = {2023},
month = nov,
journal = {Nonlinearity},
volume = {36},
number = {11},
eprint = {2211.00450},
primaryclass = {math, stat},
pages = {5731--5772},
issn = {0951-7715, 1361-6544},
doi = {10.1088/1361-6544/acf988},
urldate = {2023-12-25},
archiveprefix = {arXiv},
keywords = {Mathematics - Analysis of PDEs,Mathematics - Probability,Statistics - Machine Learning}
}
@article{mielke2025hellinger,
title = {{Hellinger-Kantorovich} gradient flows: Global exponential decay of entropy functionals},
author = {Mielke, Alexander and Zhu, Jia-Jie},
journal = {arXiv preprint arXiv:2501.17049},
year = {2025}
}
@article{wang2022exponentially,
title = {An exponentially converging particle method for the mixed {Nash} equilibrium of continuous games},
author = {Wang, Guillaume and Chizat, L{\'e}na{\"\i}c},
journal = {arXiv preprint arXiv:2211.01280},
year = {2022}
}
@article{gaoDistributionallyRobustStochastic2016,
title = {Distributionally robust stochastic optimization with {Wasserstein} distance},
author = {Gao, Rui and Kleywegt, Anton J.},
year = {2016},
journal = {arXiv preprint arXiv:1604.02199},
}
@article{zhaoDatadrivenRiskaverseStochastic2018,
title = {Data-driven risk-averse stochastic optimization with {Wasserstein} metric},
author = {Zhao, Chaoyue and Guan, Yongpei},
year = {2018},
month = mar,
journal = {Operations Research Letters},
volume = {46},
number = {2},
pages = {262--267},
issn = {01676377},
doi = {10.1016/j.orl.2018.01.011},
urldate = {2022-03-03},
langid = {english}
}
@article{wang2021sinkhorn,
title={{Sinkhorn} distributionally robust optimization},
author={Wang, Jie and Gao, Rui and Xie, Yao},
journal={arXiv preprint arXiv:2109.11926},
year={2021}
}
@inproceedings{lee2021structured,
title={Structured logconcave sampling with a restricted {Gaussian} oracle},
author={Lee, Yin Tat and Shen, Ruoqi and Tian, Kevin},
booktitle={Conference on Learning Theory},
pages={2993--3050},
year={2021},
organization={PMLR}
}
@article{sinha2020certifyingdistributionalrobustnessprincipled,
title={Certifying some distributional robustness with principled adversarial training},
author={Sinha, Aman and Namkoong, Hongseok and Volpi, Riccardo and Duchi, John},
journal={arXiv preprint arXiv:1710.10571},
year={2017}
}
@inproceedings{chen2022improved,
title={Improved analysis for a proximal algorithm for sampling},
author={Chen, Yongxin and Chewi, Sinho and Salim, Adil and Wibisono, Andre},
booktitle={Conference on Learning Theory},
pages={2984--3014},
year={2022},
organization={PMLR}
}
@article{bertsimas2020predictive,
title={From predictive to prescriptive analytics},
author={Bertsimas, Dimitris and Kallus, Nathan},
journal={Management Science},
volume={66},
number={3},
pages={1025--1044},
year={2020},
publisher={INFORMS}
}
@article{chewi2024statistical,
title={Statistical optimal transport},
author={Chewi, Sinho and Niles-Weed, Jonathan and Rigollet, Philippe},
journal={arXiv preprint arXiv:2407.18163},
year={2024}
}
@article{xu2018global,
title={Global convergence of {Langevin} dynamics based algorithms for nonconvex optimization},
author={Xu, Pan and Chen, Jinghui and Zou, Difan and Gu, Quanquan},
journal={Advances in Neural Information Processing Systems},
volume={31},
year={2018}
}
@article{hwang1980laplace,
title={{Laplace's} method revisited: weak convergence of probability measures},
author={Hwang, Chii-Ruey},
journal={The Annals of Probability},
pages={1177--1182},
year={1980},
publisher={JSTOR}
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@article{gelfand1991recursive,
title={Recursive stochastic algorithms for global optimization in {R}\^{}d},
author={Gelfand, Saul B and Mitter, Sanjoy K},
journal={SIAM Journal on Control and Optimization},
volume={29},
number={5},
pages={999--1018},
year={1991},
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}
@article{roberts1996exponential,
title={Exponential convergence of {Langevin} distributions and their discrete approximations},
author={Roberts, Gareth O and Tweedie, Richard L},
year={1996}
}
@article{tan2023accelerate,
title={Accelerate {Langevin} sampling with birth-death process and exploration component},
author={Tan, Lezhi and Lu, Jianfeng},
journal={arXiv preprint arXiv:2305.05529},
year={2023}
}
@article{mohajerin2018data,
title={Data-driven distributionally robust optimization using the {Wasserstein} metric: Performance guarantees and tractable reformulations},
author={Mohajerin Esfahani, Peyman and Kuhn, Daniel},
journal={Mathematical Programming},
volume={171},
number={1},
pages={115--166},
year={2018},
publisher={Springer}
}
@article{nusken2024stein,
title={{Stein} transport for {Bayesian} inference},
author={N{\"u}sken, Nikolas},
journal={arXiv preprint arXiv:2409.01464},
year={2024}
}
@article{liu2016stein,
title={{Stein} variational gradient descent: A general purpose {Bayesian} inference algorithm},
author={Liu, Qiang and Wang, Dilin},
journal={Advances in neural information processing systems},
volume={29},
year={2016}
}
@article{heng2024diffusion,
title={Diffusion {Schr{\"o}dinger} bridges for {Bayesian} computation},
author={Heng, Jeremy and De Bortoli, Valentin and Doucet, Arnaud},
journal={Statistical Science},
volume={39},
number={1},
pages={90--99},
year={2024},
publisher={Institute of Mathematical Statistics}
}
@article{hu2013kullback,
title={{Kullback-Leibler} divergence constrained distributionally robust optimization},
author={Hu, Zhaolin and Hong, L Jeff},
journal={Available at Optimization Online},
volume={1},
number={2},
pages={9},
year={2013}
}
@article{maurais2024sampling,
title={Sampling in unit time with kernel {Fisher-Rao} flow},
author={Maurais, Aimee and Marzouk, Youssef},
journal={arXiv preprint arXiv:2401.03892},
year={2024}
}
@article{ghadimi2013stochastic,
title={Stochastic first-and zeroth-order methods for nonconvex stochastic programming},
author={Ghadimi, Saeed and Lan, Guanghui},
journal={SIAM journal on optimization},
volume={23},
number={4},
pages={2341--2368},
year={2013},
publisher={SIAM}
}
@article{vempala2019rapid,
title={Rapid convergence of the unadjusted {Langevin} algorithm: Isoperimetry suffices},
author={Vempala, Santosh and Wibisono, Andre},
journal={Advances in neural information processing systems},
volume={32},
year={2019}
}
@article{talagrand1996transportation,
title={Transportation cost for {Gaussian} and other product measures},
author={Talagrand, Michel},
journal={Geometric \& Functional Analysis GAFA},
volume={6},
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publisher={Springer}
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@article{zhu2024distributionally,
title={Distributionally robust optimization via iterative algorithms in continuous probability spaces},
author={Zhu, Linglingzhi and Xie, Yao},
journal={arXiv preprint arXiv:2412.20556},
year={2024}
}
@article{xu2024flow,
title={Flow-based distributionally robust optimization},
author={Xu, Chen and Lee, Jonghyeok and Cheng, Xiuyuan and Xie, Yao},
journal={IEEE Journal on Selected Areas in Information Theory},
volume={5},
pages={62--77},
year={2024},
publisher={IEEE}
}
@inproceedings{zhu2021kernel,
title={Kernel distributionally robust optimization: Generalized duality theorem and stochastic approximation},
author={Zhu, Jia-Jie and Jitkrittum, Wittawat and Diehl, Moritz and Sch{\"o}lkopf, Bernhard},
booktitle={International Conference on Artificial Intelligence and Statistics},
pages={280--288},
year={2021},
organization={PMLR}
}
@article{el1997robust,
title={Robust solutions to least-squares problems with uncertain data},
author={El Ghaoui, Laurent and Lebret, Herv{\'e}},
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@misc{krizhevsky2009learning,
title={Learning multiple layers of features from tiny images.(2009)},
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year={2009}
}
@book{otto1996double,
title={Double degenerate diffusion equations as steepest descent},
author={Otto, Felix},
year={1996},
publisher={Sonderforschungsbereich 256}
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@article{garcia2018continuum,
title={Continuum limits of posteriors in graph {Bayesian} inverse problems},
author={Garc{\'\i}a Trillos, Nicolás and Sanz-Alonso, Daniel},
journal={SIAM Journal on Mathematical Analysis},
volume={50},
number={4},
pages={4020--4040},
year={2018},
publisher={SIAM}
}
@article{onsagerFluctuationsIrreversibleProcesses1953,
title={Fluctuations and irreversible processes},
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title={Reciprocal relations in irreversible processes. I.},
author={Onsager, Lars},
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publisher={APS}
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@article{mielkeNonequilibriumThermodynamicalPrinciples2017,
title={Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics},
author={Mielke, Alexander and Patterson, Robert IA and Peletier, Mark A and Michiel Renger, DR3691731},
journal={SIAM Journal on Applied Mathematics},
volume={77},
number={4},
pages={1562--1585},
year={2017},
publisher={SIAM}
}
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title={The geometry of dissipative evolution equations: the porous medium equation},
author={Otto, Felix},
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@article{mielke2023introduction,
title={An introduction to the analysis of gradients systems},
author={Mielke, Alexander},
journal={arXiv preprint arXiv:2306.05026},
year={2023}
}
@article{wibisono2025mixing,
title={Mixing time of the proximal sampler in relative {Fisher} information via strong data processing inequality},
author={Wibisono, Andre},
journal={arXiv preprint arXiv:2502.05623},
year={2025}
}
@inproceedings{chewi2022query,
title={The query complexity of sampling from strongly log-concave distributions in one dimension},
author={Chewi, Sinho and Gerber, Patrik R and Lu, Chen and Le Gouic, Thibaut and Rigollet, Philippe},
booktitle={Conference on Learning Theory},
pages={2041--2059},
year={2022},
organization={PMLR}
}
@inproceedings{ba2021understanding,
title={Understanding the variance collapse of {SVGD} in high dimensions},
author={Ba, Jimmy and Erdogdu, Murat A and Ghassemi, Marzyeh and Sun, Shengyang and Suzuki, Taiji and Wu, Denny and Zhang, Tianzong},
booktitle={International Conference on Learning Representations},
year={2021}
}
@inproceedings{salim2022convergence,
title={A convergence theory for {SVGD} in the population limit under {Talagrand's} inequality {T1}},
author={Salim, Adil and Sun, Lukang and Richtarik, Peter},
booktitle={International Conference on Machine Learning},
pages={19139--19152},
year={2022},
organization={PMLR}
}
@inproceedings{zhuo2018message,
title={Message passing {Stein} variational gradient descent},
author={Zhuo, Jingwei and Liu, Chang and Shi, Jiaxin and Zhu, Jun and Chen, Ning and Zhang, Bo},
booktitle={International Conference on Machine Learning},
pages={6018--6027},
year={2018},
organization={PMLR}
}
@article{shi2023finite,
title={A finite-particle convergence rate for {Stein} variational gradient descent},
author={Shi, Jiaxin and Mackey, Lester},
journal={Advances in Neural Information Processing Systems},
volume={36},
pages={26831--26844},
year={2023}
}
@article{gross1975logarithmic,
title={Logarithmic {Sobolev} inequalities},
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@inproceedings{blanchet2015unbiased,
title={Unbiased {Monte Carlo} for optimization and functions of expectations via multi-level randomization},
author={Blanchet, Jose H and Glynn, Peter W},
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pages={3656--3667},
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organization={IEEE}
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@inproceedings{pidhorskyi2020adversarial,
title={Adversarial latent autoencoders},
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booktitle={Proceedings of the IEEE/CVF conference on computer vision and pattern recognition},
pages={14104--14113},
year={2020}
}
@article{lecun2002gradient,
title={Gradient-based learning applied to document recognition},
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@article{jordan1998variational,
title={The variational formulation of the {Fokker--Planck} equation},
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@article{otto2000generalization,
title={Generalization of an inequality by {Talagrand} and links with the logarithmic {Sobolev} inequality},
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@article{peyre2015entropic,
title={Entropic approximation of {Wasserstein} gradient flows},
author={Peyr{\'e}, Gabriel},
journal={SIAM Journal on Imaging Sciences},
volume={8},
number={4},
pages={2323--2351},
year={2015},
publisher={SIAM}
}
@article{wang2025iterative,
title={Iterative sampling methods for {Sinkhorn} distributionally robust optimization},
author={Wang, Jie},
journal={arXiv preprint arXiv:2512.12550},
year={2025}
}
@article{dapogny2023entropy,
title={Entropy-regularized {Wasserstein} distributionally robust shape and topology optimization},
author={Dapogny, Charles and Iutzeler, Franck and Meda, Andrea and Thibert, Boris},
journal={Structural and Multidisciplinary Optimization},
volume={66},
number={3},
pages={42},
year={2023},
publisher={Springer}
}
@article{nemirovski2009robust,
title={Robust stochastic approximation approach to stochastic programming},
author={Nemirovski, Arkadi and Juditsky, Anatoli and Lan, Guanghui and Shapiro, Alexander},
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@article{madry2017towards,
title={Towards deep learning models resistant to adversarial attacks},
author={Madry, Aleksander and Makelov, Aleksandar and Schmidt, Ludwig and Tsipras, Dimitris and Vladu, Adrian},
journal={arXiv preprint arXiv:1706.06083},
year={2017}
}
@article{johnston2025performance,
title={The performance of the unadjusted {Langevin} algorithm without smoothness assumptions},
author={Johnston, Tim and Lytras, Iosif and Makras, Nikolaos and Sabanis, Sotirios},
journal={arXiv preprint arXiv:2502.03458},
year={2025}
}
@inproceedings{kantorovich1942translocation,
title={On the translocation of masses},
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