Temporary references for the minicourse.¶
# References by Section
Generated from citation commands in main.tex and the included sections/*.tex files. References are deduplicated within each section in first-use order.
## 1. Generative Tasks
Bogachev, Vladimir I. “Measure theory.” Springer, vol. 1. 2007. [bogachev2007measure]
## 2. Learning by Neural Networks
Goodfellow, Ian, Bengio, Yoshua, Courville, Aaron. “Deep learning.” MIT press. 2016. [goodfellow2016deep]
Wibisono, Andre. “Gradient flow and gradient descent.” 2016. URL: https://awibisono.github.io/2016/06/13/gradient-flow-gradient-descent.html (accessed 2026-05-29). [wibisono2016gradient]
Cybenko, George. “Approximation by superpositions of a sigmoidal function.” Mathematics of control, signals and systems, vol. 2, no. 4, pp. 303–314. Springer. 1989. [cybenko1989approximation]
Kingma, Diederik P, Ba, Jimmy. “Adam: A method for stochastic optimization.” arXiv preprint arXiv:1412.6980. 2014. [kingma2014adam]
## 3. Metrics on Probability Spaces
Levin, David A, Peres, Yuval. “Markov chains and mixing times.” American Mathematical Soc. 2017. [levin2017markov]
Conforti, Giovanni. “Weak semiconvexity estimates for Schrödinger potentials and logarithmic Sobolev inequality for Schrödinger bridges.” arXiv preprint arXiv:2301.00083. 2022. [conforti2022weak]
Bogachev, Vladimir I. “Measure theory.” Springer, vol. 1. 2007. [bogachev2007measure]
Villani, Cédric. “Optimal transport: old and new.” Springer, vol. 338. 2009. [villani2009optimal]
Ambrosio, Luigi, Gigli, Nicola, Savaré, Giuseppe. “Gradient flows: in metric spaces and in the space of probability measures.” Springer. 2005. [ambrosio2005gradient]
Arjovsky, Martin, Chintala, Soumith, Bottou, Léon. “Wasserstein generative adversarial networks.” International Conference on Machine Learning, pp. 214–223. PMLR. 2017. [arjovsky2017wasserstein]
## 4. Generative Adversarial Networks
Goodfellow, Ian, Pouget-Abadie, Jean, Mirza, Mehdi, Xu, Bing, Warde-Farley, David, Ozair, Sherjil, Courville, Aaron, Bengio, Yoshua. “Generative adversarial nets.” Advances in Neural Information Processing Systems, vol. 27. 2014. [goodfellow2014generative]
Goodfellow, Ian, Bengio, Yoshua, Courville, Aaron. “Deep learning.” MIT press. 2016. [goodfellow2016deep]
Arjovsky, Martin, Chintala, Soumith, Bottou, Léon. “Wasserstein generative adversarial networks.” International Conference on Machine Learning, pp. 214–223. PMLR. 2017. [arjovsky2017wasserstein]
Gulrajani, Ishaan, Ahmed, Faruk, Arjovsky, Martin, Dumoulin, Vincent, Courville, Aaron. “Improved Training of Wasserstein GANs.” Advances in Neural Information Processing Systems, vol. 30. 2017. [gulrajani2017improved]
Miyato, Takeru, Kataoka, Toshiki, Koyama, Masanori, Yoshida, Yuichi. “Spectral Normalization for Generative Adversarial Networks.” International Conference on Learning Representations. 2018. [miyato2018spectral]
Xie, Yue, Wang, Zhongjian, Zhang, Zhiwen. “Randomized methods for computing optimal transport without regularization and their convergence analysis.” Journal of Scientific Computing, vol. 100, no. 2, pp. 37. Springer. 2024. [xie2024randomized]
Amos, Brandon. “On Amortizing Convex Conjugates for Optimal Transport.” International Conference on Learning Representations. 2023. [amos2023amortizing]
Villani, Cédric. “Optimal transport: old and new.” Springer, vol. 338. 2009. [villani2009optimal]
Wang, Zhongjian, Xin, Jack, Zhang, Zhiwen. “DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle method.” Journal of Computational Physics, vol. 464, pp. 111309. Elsevier. 2022. [wang2022deepparticle]
Wang, Zhongjian, Xin, Jack, Zhang, Zhiwen. “A DeepParticle method for learning and generating aggregation patterns in multi-dimensional Keller-Segel chemotaxis systems.” Physica D: Nonlinear Phenomena, vol. 460, pp. 134082. Elsevier. 2024. [wang2024deepparticle_physicad]
Zhang, Tian, Wang, Zhongjian, Xin, Jack, Zhang, Zhiwen. “A bidirectional DeepParticle method for efficiently solving low-dimensional transport map problems.” Journal of Computational Physics. Elsevier. 2026. [zhang2026bidirectional]
Makkuva, Ashok, Taghvaei, Amirhossein, Oh, Sewoong, Lee, Jason. “Optimal transport mapping via input convex neural networks.” International Conference on Machine Learning, pp. 6672–6681. PMLR. 2020. [makkuva2020optimal]
Korotin, Alexander, Egiazarian, Vage, Asadulaev, Arip, Safin, Alexander, Burnaev, Evgeny. “Wasserstein-2 generative networks.” arXiv preprint arXiv:1909.13082. 2019. [korotin2019wasserstein]
Li, Yingyuan, Wang, Aokun, Wang, Zhongjian. “DPOT: A DeepParticle method for Computation of Optimal Transport with convergence guarantee.” arXiv preprint arXiv:2506.23429. 2025. [li2025dpot]
Song, Fanghui, Wang, Zhongjian, Sun, Jiebao. “Physics-Informed Design of Input Convex Neural Networks for Consistency Optimal Transport Flow Matching.” arXiv preprint arXiv:2511.06042. 2025. [song2025physics]
Kornilov, Nikita, Mokrov, Petr, Gasnikov, Alexander, Korotin, Alexander. “Optimal Flow Matching: Learning Straight Trajectories in Just One Step.” arXiv preprint arXiv:2403.13117. 2024. [kornilov2024optimal_flow_matching]
## 5. Normalizing Flows
Dinh, Laurent, Sohl-Dickstein, Jascha, Bengio, Samy. “Density estimation using Real NVP.” International Conference on Learning Representations. 2017. [dinh2016density]
Tang, Kejun, Wan, Xiaoliang, Liao, Qifeng. “KRnet: A neural network implementation of the Knothe-Rosenblatt rearrangement.” Journal of Computational Physics, vol. 430, pp. 110098. Elsevier. 2021. [tang2021krnet]
Papamakarios, George, Nalisnick, Eric, Rezende, Danilo Jimenez, Mohamed, Shakir, Lakshminarayanan, Balaji. “Normalizing flows for probabilistic modeling and inference.” Journal of Machine Learning Research, vol. 22, no. 57, pp. 1–64. 2021. [papamakarios2021normalizing]
Rezende, Danilo Jimenez, Mohamed, Shakir. “Variational inference with normalizing flows.” International Conference on Machine Learning, pp. 1530–1538. PMLR. 2015. [rezende2015variational]
Stuart, Andrew M. “Inverse problems: a Bayesian perspective.” Acta numerica, vol. 19, pp. 451–559. Cambridge University Press. 2010. [stuart2010inverse]
Goodfellow, Ian, Pouget-Abadie, Jean, Mirza, Mehdi, Xu, Bing, Warde-Farley, David, Ozair, Sherjil, Courville, Aaron, Bengio, Yoshua. “Generative adversarial nets.” Advances in Neural Information Processing Systems, vol. 27. 2014. [goodfellow2014generative]
Arjovsky, Martin, Chintala, Soumith, Bottou, Léon. “Wasserstein generative adversarial networks.” International Conference on Machine Learning, pp. 214–223. PMLR. 2017. [arjovsky2017wasserstein]
Wang, Zhongjian, Xin, Jack, Zhang, Zhiwen. “DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle method.” Journal of Computational Physics, vol. 464, pp. 111309. Elsevier. 2022. [wang2022deepparticle]
## 6. SDE Preliminaries
Song, Yang, Sohl-Dickstein, Jascha, Kingma, Diederik P, Kumar, Arvind, Ermon, Stefano, Poole, Ben. “Score-Based Generative Modeling through Stochastic Differential Equations.” International Conference on Learning Representations. 2021. [song2021score]
Ho, Jonathan, Jain, Ajay, Abbeel, Pieter. “Denoising diffusion probabilistic models.” Advances in Neural Information Processing Systems, vol. 33, pp. 6840–6851. 2020. [ho2020denoising]
Øksendal, Bernt. “Stochastic differential equations.” Springer. 2003. [oksendal2003stochastic]
Pavliotis, Grigorios A. “Stochastic processes and applications: diffusion processes, the Fokker-Planck and Langevin equations.” Springer. 2014. [pavliotis2014stochastic]
## 7. Numerical Solution of SDEs
Kloeden, Peter E, Platen, Eckhard. “Numerical solution of stochastic differential equations.” Springer, vol. 23. 1992. [kloeden1992numerical]
Øksendal, Bernt. “Stochastic differential equations.” Springer. 2003. [oksendal2003stochastic]
Pavliotis, Grigorios A. “Stochastic processes and applications: diffusion processes, the Fokker-Planck and Langevin equations.” Springer. 2014. [pavliotis2014stochastic]
## 8. OU semigroup and Langevin Sampling
Bakry, Dominique, Gentil, Ivan, Ledoux, Michel. “Analysis and Geometry of Markov Diffusion Operators.” Grundlehren der mathematischen Wissenschaften, vol. 348. Springer. 2014. [bakry2014analysis]
Gross, Leonard. “Logarithmic Sobolev Inequalities.” American Journal of Mathematics, vol. 97, no. 4, pp. 1061–1083. 1975. DOI: 10.2307/2373688. [gross1975logarithmic]
## 9. OU Noising and Langevin Sampling
Dalalyan, Arnak S. “Theoretical guarantees for approximate sampling from smooth and log-concave densities.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol. 79, no. 3, pp. 651–676. Wiley. 2017. [dalalyan2017theoretical]
Durmus, Alain, Moulines, Eric. “Nonasymptotic convergence analysis for the unadjusted Langevin algorithm.” The Annals of Applied Probability, vol. 27, no. 3, pp. 1551–1587. Institute of Mathematical Statistics. 2017. [durmus2017nonasymptotic]
Roberts, Gareth O, Tweedie, Richard L. “Exponential convergence of Langevin distributions and their discrete approximations.” Bernoulli, pp. 341–363. JSTOR. 1996. [roberts1996exponential]
Welling, Max, Teh, Yee Whye. “Bayesian learning via stochastic gradient Langevin dynamics.” Proceedings of the 28th international conference on machine learning (ICML-11), pp. 681–688. 2011. [welling2011bayesian]
Raginsky, Maxim, Rakhlin, Alexander, Telgarsky, Matus. “Non-convex learning via stochastic gradient Langevin dynamics: A nonasymptotic analysis.” Conference on Learning Theory, pp. 1674–1703. PMLR. 2017. [raginsky2017nonconvex]
Øksendal, Bernt. “Stochastic differential equations.” Springer. 2003. [oksendal2003stochastic]
Gross, Leonard. “Logarithmic Sobolev Inequalities.” American Journal of Mathematics, vol. 97, no. 4, pp. 1061–1083. 1975. DOI: 10.2307/2373688. [gross1975logarithmic]
Bakry, Dominique, Gentil, Ivan, Ledoux, Michel. “Analysis and Geometry of Markov Diffusion Operators.” Grundlehren der mathematischen Wissenschaften, vol. 348. Springer. 2014. [bakry2014analysis]
## 10. Construction by Reversed Heat Flow
Anderson, Brian DO. “Reverse-time diffusion equation models.” Stochastic Processes and their Applications, vol. 12, no. 3, pp. 313–326. Elsevier. 1982. [anderson1982reverse]
Song, Yang, Sohl-Dickstein, Jascha, Kingma, Diederik P, Kumar, Arvind, Ermon, Stefano, Poole, Ben. “Score-Based Generative Modeling through Stochastic Differential Equations.” International Conference on Learning Representations. 2021. [song2021score]
Kim, Young-Heon, Milman, Emanuel. “A generalization of Caffarelli’s contraction theorem via (reverse) heat flow.” Mathematische Annalen, vol. 354, no. 3, pp. 827–862. Springer. 2012. [kim2012generalization]
Hyvärinen, Aapo. “Estimation of non-normalized statistical models by score matching.” Journal of Machine Learning Research, vol. 6, pp. 695–709. 2005. [hyvarinen2005estimation]
Vincent, Pascal. “A connection between score matching and denoising autoencoders.” Neural Computation, vol. 23, no. 7, pp. 1661–1674. MIT Press. 2011. [vincent2011connection]
Efron, Bradley. “Tweedie’s formula and selection bias.” Journal of the American Statistical Association, vol. 106, no. 496, pp. 1602–1614. Taylor & Francis. 2011. [efron2011tweedie]
Ho, Jonathan, Jain, Ajay, Abbeel, Pieter. “Denoising diffusion probabilistic models.” Advances in Neural Information Processing Systems, vol. 33, pp. 6840–6851. 2020. [ho2020denoising]
Lu, Yubin, Wang, Zhongjian, Bal, Guillaume. “Mathematical analysis of singularities in the diffusion model under the submanifold assumption.” arXiv preprint arXiv:2301.07882. 2023. [lu2023mathematical]
Chen, Sitan, Chewi, Sinho, Li, Jerry, Li, Yuanzhi, Salim, Adil, Zhang, Anru R. “Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions.” International Conference on Learning Representations. 2023. [chen2023sampling]
Chen, Hongrui, Lee, Holden, Lu, Jianfeng. “Improved analysis of score-based generative modeling: User-friendly bounds under minimal smoothness assumptions.” International Conference on Machine Learning, pp. 4735–4763. PMLR. 2023. [chen2023improved]
Benton, Joe, De Bortoli, Valentin, Doucet, Arnaud, Deligiannidis, George. “Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization.” International Conference on Learning Representations. 2024. [benton2024linear]
Kloeden, Peter E, Platen, Eckhard. “Numerical solution of stochastic differential equations.” Springer, vol. 23. 1992. [kloeden1992numerical]
Chen, Ricky T. Q., Rubanova, Yulia, Bettencourt, Jesse, Duvenaud, David K. “Neural ordinary differential equations.” Advances in Neural Information Processing Systems, vol. 31. 2018. [chen2018neuralode]
Liu, Xingchao, Gong, Chengyue, Liu, Qiang. “Flow straight and fast: Learning to generate and transfer data with rectified flow.” International Conference on Learning Representations. 2023. [liu2023flow]
Lipman, Yaron, Chen, Ricky TQ, Ben-Hamu, Heli, Nickel, Maximilian, Le, Matt. “Flow matching for generative modeling.” International Conference on Learning Representations. 2023. [lipman2023flow]
Song, Yang, Dhariwal, Prafulla, Chen, Mark, Sutskever, Ilya. “Consistency Models.” International Conference on Machine Learning, pp. 32211–32252. PMLR. 2023. [song2023consistency]
Meng, Xiangjun, Wang, Zhongjian. “Pathway to O(sqrt(d)) Complexity bound under Wasserstein metric of flow-based models.” arXiv preprint arXiv:2512.06702. 2025. [meng2025pathway]
Albergo, Michael S, Vanden-Eijnden, Eric. “Building normalizing flows with stochastic interpolants.” International Conference on Learning Representations. 2023. [albergo2023building]
Föllmer, Hans. “Random fields and diffusion processes.” École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87, pp. 101–203. Springer. 1988. [follmer1988random]
## 11. Bayesian Inverse Problems and Guidance
Ho, Jonathan, Salimans, Tim. “Classifier-free diffusion guidance.” NeurIPS 2021 Workshop on Deep Generative Models and Downstream Applications. 2021. [ho2022classifier]
Chung, Hyungjin, Kim, Jeongsol, McCann, Michael T, Klasky, Marc L, Ye, Jong Chul. “Diffusion posterior sampling for general noisy inverse problems.” International Conference on Learning Representations. 2023. [chung2023diffusion]
Wang, Hengkang, Zhang, Xu, Li, Taihui, Wan, Yuxiang, Chen, Tiancong, Sun, Ju. “Dmplug: A plug-in method for solving inverse problems with diffusion models.” Advances in Neural Information Processing Systems, vol. 37, pp. 117881–117916. 2024. [wang2024dmplug]
Stuart, Andrew M. “Inverse problems: a Bayesian perspective.” Acta numerica, vol. 19, pp. 451–559. Cambridge University Press. 2010. [stuart2010inverse]
## 12. Early Stopping and Quadratic Schedule
Chen, Sitan, Chewi, Sinho, Li, Jerry, Li, Yuanzhi, Salim, Adil, Zhang, Anru R. “Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions.” International Conference on Learning Representations. 2023. [chen2023sampling]
Lee, Holden, Lu, Jianfeng, Tan, Yixin. “Convergence of score-based generative modeling for general data distributions.” Proceedings of The 34th International Conference on Algorithmic Learning Theory, vol. 201, pp. 946–985. PMLR. 2023. [lee2023convergence]
Chen, Hongrui, Lee, Holden, Lu, Jianfeng. “Improved analysis of score-based generative modeling: User-friendly bounds under minimal smoothness assumptions.” International Conference on Machine Learning, pp. 4735–4763. PMLR. 2023. [chen2023improved]
Lu, Yubin, Wang, Zhongjian, Bal, Guillaume. “Mathematical analysis of singularities in the diffusion model under the submanifold assumption.” arXiv preprint arXiv:2301.07882. 2023. [lu2023mathematical]
Lyu, Zhaoyang, Xu, Xudong, Yang, Ceyuan, Lin, Dahua, Dai, Bo. “Accelerating Diffusion Models via Early Stop of the Diffusion Process.” arXiv preprint arXiv:2205.12524. 2022. [lyu2022accelerating]
## 13. 2nd Order Schemes
Karras, Tero, Aittala, Miika, Aila, Timo, Laine, Samuli. “Elucidating the design space of diffusion-based generative models.” Advances in Neural Information Processing Systems, vol. 35, pp. 26565–26577. 2022. [karras2022elucidating]
Kloeden, Peter E, Platen, Eckhard. “Numerical solution of stochastic differential equations.” Springer, vol. 23. 1992. [kloeden1992numerical]
Salimans, Tim, Ho, Jonathan. “Progressive distillation for fast sampling of diffusion models.” International Conference on Learning Representations. 2022. [salimans2022progressive]
Song, Yang, Dhariwal, Prafulla, Chen, Mark, Sutskever, Ilya. “Consistency Models.” International Conference on Machine Learning, pp. 32211–32252. PMLR. 2023. [song2023consistency]
## 14. Distillations
Salimans, Tim, Ho, Jonathan. “Progressive distillation for fast sampling of diffusion models.” International Conference on Learning Representations. 2022. [salimans2022progressive]
Song, Yang, Dhariwal, Prafulla, Chen, Mark, Sutskever, Ilya. “Consistency Models.” International Conference on Machine Learning, pp. 32211–32252. PMLR. 2023. [song2023consistency]
Frans, Kevin, Hafner, Danijar, Levine, Sergey, Abbeel, Pieter. “One step diffusion via shortcut models.” International Conference on Learning Representations, vol. 2025, pp. 34668–34684. 2025. [frans2025one]
Liu, Xingchao, Gong, Chengyue, Liu, Qiang. “Flow straight and fast: Learning to generate and transfer data with rectified flow.” International Conference on Learning Representations. 2023. [liu2023flow]
Geng, Zhengyang, Deng, Mingyang, Bai, Xingjian, Kolter, J Zico, He, Kaiming. “Mean flows for one-step generative modeling.” Advances in Neural Information Processing Systems. 2025. [geng2025mean]
Léonard, Christian. “From the Schrödinger problem to the Monge–Kantorovich problem.” Journal of Functional Analysis, vol. 262, no. 4, pp. 1879–1920. 2012. DOI: 10.1016/j.jfa.2011.11.002. [leonard2012from]
Léonard, Christian. “A survey of the Schrödinger problem and some of its connections with optimal transport.” Discrete and Continuous Dynamical Systems. Series A, vol. 34, no. 4, pp. 1533–1574. 2014. DOI: 10.3934/dcds.2014.34.1533. [leonard2014survey]
Gentiloni Silveri, Marta, Conforti, Giovanni, Durmus, Alain. “Exponential Convergence Guarantees for Iterative Markovian Fitting.” arXiv preprint arXiv:2510.20871. 2025. DOI: 10.48550/arXiv.2510.20871. [gentiloni2025exponential]
## 15. Two-Step Hybrid Models
Shen, Zhenda, Wang, Zhongjian, Xin, Jack, Zhang, Zhiwen. “Two-step diffusion: fast sampling and reliable prediction for 3D Keller-Segel and KPP equations in fluid flows.” arXiv preprint arXiv:2601.20024. 2026. [shen2026two]
Song, Fanghui, Wang, Zhongjian, Sun, Jiebao. “Physics-Informed Design of Input Convex Neural Networks for Consistency Optimal Transport Flow Matching.” arXiv preprint arXiv:2511.06042. 2025. [song2025physics]
Wang, Hengkang, Zhang, Xu, Li, Taihui, Wan, Yuxiang, Chen, Tiancong, Sun, Ju. “Dmplug: A plug-in method for solving inverse problems with diffusion models.” Advances in Neural Information Processing Systems, vol. 37, pp. 117881–117916. 2024. [wang2024dmplug]
Zhang, Tian, Wang, Zhongjian, Xin, Jack, Zhang, Zhiwen. “A bidirectional DeepParticle method for efficiently solving low-dimensional transport map problems.” Journal of Computational Physics. Elsevier. 2026. [zhang2026bidirectional]
Wang, Aokun, Nair, Anjali, Wang, Zhongjian, Bal, Guillaume. “ML-based approach to classification and generation of structured light propagation in turbulent media.” arXiv preprint arXiv:2604.14208. 2026. [wang2026ml]
## 16. Numerical analysis of SDE
Chen, Sitan, Chewi, Sinho, Li, Jerry, Li, Yuanzhi, Salim, Adil, Zhang, Anru R. “Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions.” International Conference on Learning Representations. 2023. [chen2023sampling]
Benton, Joe, De Bortoli, Valentin, Doucet, Arnaud, Deligiannidis, George. “Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization.” International Conference on Learning Representations. 2024. [benton2024linear]
Conforti, Giovanni. “Weak semiconvexity estimates for Schrödinger potentials and logarithmic Sobolev inequality for Schrödinger bridges.” arXiv preprint arXiv:2301.00083. 2022. [conforti2022weak]
Chen, Hongrui, Lee, Holden, Lu, Jianfeng. “Improved analysis of score-based generative modeling: User-friendly bounds under minimal smoothness assumptions.” International Conference on Machine Learning, pp. 4735–4763. PMLR. 2023. [chen2023improved]
Kloeden, Peter E, Platen, Eckhard. “Numerical solution of stochastic differential equations.” Springer, vol. 23. 1992. [kloeden1992numerical]
Wang, Xixian, Wang, Zhongjian. “Wasserstein bounds for generative diffusion models with Gaussian tail targets.” arXiv preprint arXiv:2412.11251. 2024. [wang2024wasserstein]
Meng, Xiangjun, Wang, Zhongjian. “Pathway to O(sqrt(d)) Complexity bound under Wasserstein metric of flow-based models.” arXiv preprint arXiv:2512.06702. 2025. [meng2025pathway]
## 17. Lipschitzness of score
Mooney, Connor, Wang, Zhongjian, Xin, Jack, Yu, Yifeng. “Global well-posedness and convergence analysis of score-based generative models via sharp Lipschitz estimates.” arXiv preprint arXiv:2405.16104. 2024. [mooney2024global]
Lin, Likun, Wang, Zhongjian. “On the Regularity and Generalization of One-Step Wasserstein-guided Generative Models for PDE-Induced Measures.” arXiv preprint arXiv:2605.21388. 2026. [lin2026regularity]
Wang, Xixian, Wang, Zhongjian. “Wasserstein bounds for generative diffusion models with Gaussian tail targets.” arXiv preprint arXiv:2412.11251. 2024. [wang2024wasserstein]
Lu, Yubin, Wang, Zhongjian, Bal, Guillaume. “Mathematical analysis of singularities in the diffusion model under the submanifold assumption.” arXiv preprint arXiv:2301.07882. 2023. [lu2023mathematical]
Meng, Xiangjun, Wang, Zhongjian. “Pathway to O(sqrt(d)) Complexity bound under Wasserstein metric of flow-based models.” arXiv preprint arXiv:2512.06702. 2025. [meng2025pathway]
Chen, Hongrui, Lee, Holden, Lu, Jianfeng. “Improved analysis of score-based generative modeling: User-friendly bounds under minimal smoothness assumptions.” International Conference on Machine Learning, pp. 4735–4763. PMLR. 2023. [chen2023improved]
Benton, Joe, De Bortoli, Valentin, Doucet, Arnaud, Deligiannidis, George. “Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization.” International Conference on Learning Representations. 2024. [benton2024linear]
## 18. Lipschitz Changes of Variables and Stability
Brenier, Yann. “Polar factorization and monotone rearrangement of vector-valued functions.” Communications on Pure and Applied Mathematics, vol. 44, no. 4, pp. 375–417. 1991. DOI: 10.1002/cpa.3160440402. [brenier1991polar]
Villani, Cédric. “Optimal transport: old and new.” Springer, vol. 338. 2009. [villani2009optimal]
Caffarelli, Luis A. “Monotonicity properties of optimal transportation and the FKG and related inequalities.” Communications in Mathematical Physics, vol. 214, pp. 547–563. Springer. 2000. [caffarelli2000monotonicity]
Kim, Young-Heon, Milman, Emanuel. “A generalization of Caffarelli’s contraction theorem via (reverse) heat flow.” Mathematische Annalen, vol. 354, no. 3, pp. 827–862. Springer. 2012. [kim2012generalization]
Fathi, Max, Mikulincer, Dan, Shenfeld, Yair. “Transportation onto log-Lipschitz perturbations.” Calculus of Variations and Partial Differential Equations, vol. 63, no. 3, pp. 61. Springer. 2024. [fathi2024transportation]
Brigati, Giovanni, Pedrotti, Francesco. “Heat flow, log-concavity, and Lipschitz transport maps.” Electronic Communications in Probability, vol. 30, pp. Paper No. 71, 12. 2025. DOI: 10.1214/25-ECP717. [brigati2025heat]
Neeman, Joe. “Lipschitz changes of variables via heat flow.” arXiv preprint arXiv:2201.03403. 2022. [neeman2022lipschitz]
Wang, Xixian, Wang, Zhongjian. “Wasserstein bounds for generative diffusion models with Gaussian tail targets.” arXiv preprint arXiv:2412.11251. 2024. [wang2024wasserstein]
Meng, Xiangjun, Wang, Zhongjian. “Pathway to O(sqrt(d)) Complexity bound under Wasserstein metric of flow-based models.” arXiv preprint arXiv:2512.06702. 2025. [meng2025pathway]
## 19. Finalizing the bound: Föllmer
Meng, Xiangjun, Wang, Zhongjian. “Pathway to O(sqrt(d)) Complexity bound under Wasserstein metric of flow-based models.” arXiv preprint arXiv:2512.06702. 2025. [meng2025pathway]
Föllmer, Hans. “Random fields and diffusion processes.” École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87, pp. 101–203. Springer. 1988. [follmer1988random]
Wang, Xixian, Wang, Zhongjian. “Wasserstein bounds for generative diffusion models with Gaussian tail targets.” arXiv preprint arXiv:2412.11251. 2024. [wang2024wasserstein]
## 20. Function-Space Posterior Samplers
Cotter, Simon L, Roberts, Gareth O, Stuart, Andrew M, White, David. “MCMC methods for functions: modifying old algorithms to make them faster.” Statistical Science, pp. 424–446. JSTOR. 2013. [cotter2013mcmc]
Stuart, Andrew M. “Inverse problems: a Bayesian perspective.” Acta numerica, vol. 19, pp. 451–559. Cambridge University Press. 2010. [stuart2010inverse]
Cheng, Zilan, Wang, Li-Lian, Wang, Zhongjian. “Preconditioned one-step generative modeling for Bayesian inverse problems in function spaces.” arXiv preprint arXiv:2603.14798. 2026. [cheng2026preconditioned]
Geng, Zhengyang, Deng, Mingyang, Bai, Xingjian, Kolter, J Zico, He, Kaiming. “Mean flows for one-step generative modeling.” Advances in Neural Information Processing Systems. 2025. [geng2025mean]
Chung, Hyungjin, Kim, Jeongsol, McCann, Michael T, Klasky, Marc L, Ye, Jong Chul. “Diffusion posterior sampling for general noisy inverse problems.” International Conference on Learning Representations. 2023. [chung2023diffusion]
Wang, Xixian, Wang, Zhongjian. “Wasserstein bounds for generative diffusion models with Gaussian tail targets.” arXiv preprint arXiv:2412.11251. 2024. [wang2024wasserstein]
## 21. Drifting Models
Deng, Mingyang, Li, He, Li, Tianhong, Du, Yilun, He, Kaiming. “Generative modeling via drifting.” arXiv preprint arXiv:2602.04770. 2026. [deng2026drifting]
Lai, Chieh-Hsin, Nguyen, Bac, Murata, Naoki, Takida, Yuhta, Uesaka, Toshimitsu, Mitsufuji, Yuki, Ermon, Stefano, Tao, Molei. “A Unified View of Score-Based and Drifting Models.” arXiv preprint arXiv:2603.07514. 2026. [lai2026unified]
Turan, Erkan, Dufour, Nicolas, Ovsjanikov, Maks. “Generative Drifting is Secretly Score Matching: A Spectral and Variational Perspective.” arXiv preprint arXiv:2603.09936. 2026. [turan2026drifting_score]