@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}
}

@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},
  publisher={SIAM}
}

@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},
  number={3},
  pages={587--600},
  year={1996},
  publisher={Springer}
}

@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}},
  journal={SIAM Journal on matrix analysis and applications},
  volume={18},
  number={4},
  pages={1035--1064},
  year={1997},
  publisher={SIAM}
}

@misc{krizhevsky2009learning,
  title={Learning multiple layers of features from tiny images.(2009)},
  author={Krizhevsky, Alex and Hinton, Geoffrey and others},
  year={2009}
}

@book{otto1996double,
  title={Double degenerate diffusion equations as steepest descent},
  author={Otto, Felix},
  year={1996},
  publisher={Sonderforschungsbereich 256}
}

@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},
  author={Onsager, Lars and Machlup, Stefan},
  journal={Physical Review},
  volume={91},
  number={6},
  pages={1505},
  year={1953},
  publisher={APS}
}

@article{onsagerReciprocalRelationsIrreversible1931,
  title={Reciprocal relations in irreversible processes. I.},
  author={Onsager, Lars},
  journal={Physical review},
  volume={37},
  number={4},
  pages={405},
  year={1931},
  publisher={APS}
}

@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}
}

@article{ottoGeometryDissipativeEvolution2001,
  title={The geometry of dissipative evolution equations: the porous medium equation},
  author={Otto, Felix},
  year={2001},
  publisher={Taylor \& Francis}
}

@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},
  author={Gross, Leonard},
  journal={American Journal of Mathematics},
  volume={97},
  number={4},
  pages={1061--1083},
  year={1975},
  publisher={JSTOR}
}

@inproceedings{blanchet2015unbiased,
  title={Unbiased {Monte Carlo} for optimization and functions of expectations via multi-level randomization},
  author={Blanchet, Jose H and Glynn, Peter W},
  booktitle={2015 Winter Simulation Conference (WSC)},
  pages={3656--3667},
  year={2015},
  organization={IEEE}
}

@inproceedings{pidhorskyi2020adversarial,
  title={Adversarial latent autoencoders},
  author={Pidhorskyi, Stanislav and Adjeroh, Donald A and Doretto, Gianfranco},
  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},
  author={LeCun, Yann and Bottou, L{\'e}on and Bengio, Yoshua and Haffner, Patrick},
  journal={Proceedings of the IEEE},
  volume={86},
  number={11},
  pages={2278--2324},
  year={2002},
  publisher={Ieee}
}

@article{jordan1998variational,
  title={The variational formulation of the {Fokker--Planck} equation},
  author={Jordan, Richard and Kinderlehrer, David and Otto, Felix},
  journal={SIAM journal on mathematical analysis},
  volume={29},
  number={1},
  pages={1--17},
  year={1998},
  publisher={SIAM}
}

@article{otto2000generalization,
  title={Generalization of an inequality by {Talagrand} and links with the logarithmic {Sobolev} inequality},
  author={Otto, Felix and Villani, C{\'e}dric},
  journal={Journal of Functional Analysis},
  volume={173},
  number={2},
  pages={361--400},
  year={2000},
  publisher={Elsevier}
}

@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},
  journal={SIAM Journal on optimization},
  volume={19},
  number={4},
  pages={1574--1609},
  year={2009},
  publisher={SIAM}
}

@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},
  author={Kantorovich, Leonid V},
  booktitle={Dokl. Akad. Nauk. USSR (NS)},
  volume={37},
  pages={199--201},
  year={1942}
}