Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
129 tokens/sec
GPT-4o
28 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Squared Wasserstein-2 Distance for Efficient Reconstruction of Stochastic Differential Equations (2401.11354v1)

Published 21 Jan 2024 in math.PR, cs.LG, and stat.ME

Abstract: We provide an analysis of the squared Wasserstein-2 ($W_2$) distance between two probability distributions associated with two stochastic differential equations (SDEs). Based on this analysis, we propose the use of a squared $W_2$ distance-based loss functions in the \textit{reconstruction} of SDEs from noisy data. To demonstrate the practicality of our Wasserstein distance-based loss functions, we performed numerical experiments that demonstrate the efficiency of our method in reconstructing SDEs that arise across a number of applications.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (41)
  1. Wasserstein generative adversarial networks. In International Conference on Machine Learning, pages 214–223. PMLR, 2017.
  2. The Wasserstein space of stochastic processes. arXiv preprint arXiv:2104.14245, 2021.
  3. On a Wasserstein-type distance between solutions to stochastic differential equations. The Annals of Applied Probability, 29(3):1609–1639, 2019. doi: 10.1214/18-aap1423.
  4. Paul C Bressloff. Stochastic Processes in Cell Biology, volume 41. Springer, 2014.
  5. Statistical inference for generative models with maximum mean discrepancy. arXiv preprint arXiv:1906.05944, 2019.
  6. Neural ordinary differential equations. In Advances in Neural Information Processing Systems, volume 31, 2018.
  7. Fast and smooth interpolation on Wasserstein space. In International Conference on Artificial Intelligence and Statistics, pages 3061–3069. PMLR, 2021.
  8. Erhan Cinlar. Probability and Stochastics. Springer Science & Business Media, 2011.
  9. An elementary proof of the triangle inequality for the Wasserstein metric. Proceedings of the American Mathematical Society, 136(1):333–339, 2008.
  10. Differentiable ranks and sorting using optimal transport. In Proceedings of the 33rd International Conference on Neural Information Processing Systems, pages 6861–6871, 2019.
  11. Reduction and reconstruction of stochastic differential equations via symmetries. Journal of Mathematical Physics, 57(12), 2016.
  12. Pot: Python optimal transport. The Journal of Machine Learning Research, 22(1):3571–3578, 2021.
  13. On the rate of convergence in Wasserstein distance of the empirical measure. Probability Theory and Related Fields, 162(3-4):707–738, 2015.
  14. Learning with a Wasserstein loss. In Advances in Neural Information Pprocessing Systems, volume 28, 2015.
  15. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.
  16. Neural jump stochastic differential equations. Advances in Neural Information Processing Systems, 32, 2019.
  17. Neural SDEs as infinite-dimensional GANs. In International Conference on Machine Learning, pages 5453–5463. PMLR, 2021.
  18. Sliced Wasserstein distance for learning Gaussian mixture models. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3427–3436, 2018.
  19. Generalized sliced Wasserstein distances. In Advances in Neural Information Processing Systems, volume 32, 2019.
  20. Scalable gradients for stochastic differential equations. In International Conference on Artificial Intelligence and Statistics, pages 3870–3882. PMLR, 2020.
  21. Generative moment matching networks. In International Conference on Machine Learning, pages 1718–1727. PMLR, 2015.
  22. Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic Markov processes. Journal of The Royal Society Interface, 15(138):20170804, 2018.
  23. When Gaussian process meets big data: a review of scalable GPs. IEEE Transactions on Neural Networks and Learning Systems, 31(11):4405–4423, 2020.
  24. David JC MacKay et al. Introduction to Gaussian processes. NATO ASI Series F Computer and Systems Sciences, 168:133–166, 1998.
  25. Stochastic approaches to uncertainty quantification in CFD simulations. Numerical Algorithms, 38:209–236, 2005.
  26. Martingale Methods in Financial Modelling, volume 36. Springer Science & Business Media, 2006.
  27. Kernel Wasserstein distance. arXiv preprint arXiv:1905.09314, 2019.
  28. Learning networks of stochastic differential equations. In Advances in Neural Information Processing Systems, volume 23, 2010.
  29. Orthogonal estimation of Wasserstein distances. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 186–195. PMLR, 2019.
  30. Ludger Rüschendorf. The Wasserstein distance and approximation theorems. Probability Theory and Related Fields, 70(1):117–129, 1985.
  31. Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations. The Journal of Machine Learning Research, 22(1):11006–11042, 2021.
  32. Christian Soize. Uncertainty Quantification. Springer, 2017.
  33. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2020.
  34. Timothy John Sullivan. Introduction to Uncertainty Quantification, volume 63. Springer, 2015.
  35. Learning fractional white noises in neural stochastic differential equations. In Advances in Neural Information Processing Systems, volume 35, pages 37660–37675, 2022.
  36. Neural stochastic differential equations: deep latent Gaussian models in the diffusion limit. arXiv preprint arXiv:1905.09883, 2019.
  37. Cédric Villani et al. Optimal transport: old and new, volume 338. Springer, 2009.
  38. Jian Wang. Lpsuperscript𝐿𝑝{L}^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-Wasserstein distance for stochastic differential equations driven by Lévy processes. Bernoulli, pages 1598–1616, 2016.
  39. An introduction to the Kalman filter. 1995.
  40. Gregory F Welch. Kalman filter. Computer Vision: A Reference Guide, pages 1–3, 2020.
  41. Nonparametric different-feature selection using Wasserstein distance. In 2020 IEEE 32nd International Conference on Tools with Artificial Intelligence (ICTAI), pages 982–988. IEEE, 2020.
Citations (2)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets