Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
DE
EN
Books
Journals
Events
Topic Page
Marketing
Log in
Learn more about individual access
Request company access
Electric Powertrains and Energy Systems 2025 | 26 + 27 March 2025

19th International MTZ Congress on Future Powertrains

Take this opportunity to attend the conference. Experts from the automotive industry, energy providers, and grid operators will meet up with representatives from politics and science to discuss important topics for this sector.
EXTENDED SEARCH
Log in
Top
Published in:
Read chapter Read first chapter

2024 | OriginalPaper | Chapter

Minimum Kernel Discrepancy Estimators

Author : Chris J. Oates

Published in: Monte Carlo and Quasi-Monte Carlo Methods

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

For two decades, reproducing kernels and their associated discrepancies have facilitated elegant theoretical analyses in the setting of quasi Monte Carlo. These same tools are now receiving interest in statistics and related fields, as criteria that can be used to select an appropriate statistical model for a given dataset. The focus of this article is on minimum kernel discrepancy estimators, whose use in statistical applications is reviewed, and a general theoretical framework for establishing their asymptotic properties is presented.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Optimal Algorithms for Numerical Integration: Recent Results and Open Problems
next chapter Error Estimates and Variance Reduction for Nonequilibrium Stochastic Dynamics
Appendix
Available only for authorised users
Footnotes
1
i.e. additional levels of differentiability in directions that are axis-aligned.
 
Literature
1.
Akaike, H.: Information theory and an extension of the likelihood principle. In: Proceedings of the Second International Symposium of Information Theory (1973)
2.
Aliprantis, C.D., Burkinshaw, O.: Principles of Real Analysis. Academic Press (1998)
3.
Alquier, P., Gerber, M.: Universal robust regression via maximum mean discrepancy. Biometrika (2023). To appear
4.
Anastasiou, A., Barp, A., Briol, F.X., Ebner, B., Gaunt, R.E., Ghaderinezhad, F., Gorham, J., Gretton, A., Ley, C., Liu, Q., Mackey, L., Oates, C.J., Reinert, G., Swan, Y.: Stein’s method meets statistics: a review of some recent developments. Stat. Sci. 38(1), 120–139 (2023)MathSciNetCrossRef
5.
Arjovsky, M., Chintala, S., Bottou, L.: Wasserstein generative adversarial networks. In: Proceedings of the 34th International Conference on Machine Learning (2017)
6.
Barp, A., Briol, F.X., Duncan, A., Girolami, M., Mackey, L.: Minimum Stein discrepancy estimators. In: Proceedings of the 33rd Conference on Neural Information Processing Systems (2019)
7.
Barp, A., Simon-Gabriel, C.J., Girolami, M., Mackey, L.: Targeted separation and convergence with kernel discrepancies (2022). arXiv:​2209.​12835
8.
Basu, A., Shioya, H., Park, C.: Statistical Inference: The Minimum Distance Approach. CRC Press (2011)
9.
10.
Billingsley, P.: Probability and Measure. Wiley (1979)
11.
Bińkowski, M., Sutherland, D.J., Arbel, M., Gretton, A.: Demystifying MMD GANs. In: Proceedings of the 6th International Conference on Learning Representations (2018)
12.
Bonner, N., Kirschner, H.P.: Note on conditions for weak convergence of von Mises’ differentiable statistical functions. Ann. Stat. 5(2), 405–407 (1977)MathSciNetCrossRef
13.
Briol, F.X., Barp, A., Duncan, A.B., Girolami, M.: Statistical inference for generative models with maximum mean discrepancy (2019). arXiv:​1906.​05944
14.
Carmeli, C., De Vito, E., Toigo, A.: Vector valued reproducing kernel Hilbert spaces of integrable functions and Mercer theorem. Anal. Appl. 4(04), 377–408 (2006)MathSciNetCrossRef
15.
Chérief-Abdellatif, B.E., Alquier, P.: MMD-Bayes: robust Bayesian estimation via maximum mean discrepancy. In: Symposium on Advances in Approximate Bayesian Inference, pp. 1–21. PMLR (2020)
16.
Chérief-Abdellatif, B.E., Alquier, P.: Finite sample properties of parametric MMD estimation: robustness to misspecification and dependence. Bernoulli 28(1), 181–213 (2022)MathSciNetCrossRef
17.
Chwialkowski, K., Strathmann, H., Gretton, A.: A kernel test of goodness of fit. In: Proceedings of the 33rd International Conference on Machine Learning (2016)
18.
Cortes, E.C., Scott, C.: Sparse approximation of a kernel mean. IEEE Trans. Signal Process. 65(5), 1310–1323 (2016)MathSciNetCrossRef
19.
Davidson, J.: Stochastic Limit Theory: An Introduction for Econometricians. OUP Oxford (1994)
20.
21.
22.
Dellaporta, C., Knoblauch, J., Damoulas, T., Briol, F.X.: Robust Bayesian inference for simulator-based models via the MMD posterior bootstrap. In: Proceedings of the 25th International Conference on Artificial Intelligence and Statistics (2022)
23.
Dhariwal, P., Nichol, A.: Diffusion models beat GANs on image synthesis. In: Proceedings of the 35th Conference on Neural Information Processing Systems (2021)
24.
Dick, J., Kuo, F.Y., Sloan, I.H.: High-dimensional integration: the quasi-Monte Carlo way. Acta Numerica 22, 133–288 (2013)MathSciNetCrossRef
25.
Dick, J., Pillichshammer, F.: Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo integration. Cambridge University Press (2010)
26.
Donoho, D.L., Liu, R.C.: The “automatic’’ robustness of minimum distance functionals. Ann. Stat. 16(2), 552–586 (1988)MathSciNetCrossRef
27.
Dunford, N.: Integration of vector-valued functions. Bulletin of the American Mathematical Society, p. 43 (1937)
28.
Dziugaite, G.K., Roy, D.M., Ghahramani, Z.: Training generative neural networks via maximum mean discrepancy optimization. In: Proceedings of the 31st Conference on Uncertainty in Artificial Intelligence (2015)
29.
Frazier, D.T., Drovandi, C.: Robust approximate Bayesian inference with synthetic likelihood. J. Comput. Graph. Stat. 30(4), 958–976 (2021)MathSciNetCrossRef
30.
Freedman, D.A.: On the so-called “Huber sandwich estimator’’ and “robust standard errors’’. Am. Stat. 60(4), 299–302 (2006)MathSciNetCrossRef
31.
Genevay, A., Peyré, G., Cuturi, M.: Learning generative models with sinkhorn divergences. In: Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (2018)
32.
Gneiting, T., Raftery, A.E.: Strictly proper scoring rules, prediction, and estimation. J. Am. Stat. Assoc. 102(477), 359–378 (2007)MathSciNetCrossRef
33.
Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y.: Generative adversarial networks. Commun. ACM 63(11), 139–144 (2020)MathSciNetCrossRef
34.
Gorham, J., Mackey, L.: Measuring sample quality with kernels. In: Proceedings of the 34th International Conference on Machine Learning (2017)
35.
Gretton, A., Borgwardt, K.M., Rasch, M.J., Schölkopf, B., Smola, A.: A kernel two-sample test. J. Mach. Learn. Res. 13(1), 723–773 (2012)MathSciNet
36.
Hansen, L.P.: Large sample properties of generalized method of moments estimators. Econometrica, pp. 1029–1054 (1982)
37.
38.
Hlawka, E.: Funktionen von beschränkter variatiou in der theorie der gleichverteilung. Annali di Matematica Pura ed Applicata 54(1), 325–333 (1961)CrossRef
39.
Hoeffding, W.: A class of statistics with asymptotically normal distribution. Ann. Math. Stat. 19(3), 293–325 (1948)MathSciNetCrossRef
40.
Hoeffding, W.: The strong law of large numbers for \({U}\)-statistics. Technical report, North Carolina State University. Department of Statistics (1961)
41.
Huber, P.J.: Robust estimation of a location parameter. The Annals of Mathematical Statistics, pp. 73–101 (1964)
42.
Hyvärinen, A., Dayan, P.: Estimation of non-normalized statistical models by score matching. J. Mach. Learn. Res. 6(4) (2005)
43.
44.
Kuo, F.Y.: Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces. J. Complex. 19(3), 301–320 (2003)MathSciNetCrossRef
45.
46.
LeCun, Y., Chopra, S., Hadsell, R., Ranzato, M., Huang, F.: A tutorial on energy-based learning. In: Schölkopf, B., Smola, A.J., Taskar, B., Vishwanathan, S. (eds.), Predicting Structured Data (2007)
47.
Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer Science & Business Media (1991)
48.
Li, C.L., Chang, W.C., Cheng, Y., Yang, Y., Póczos, B.: MMD GAN: towards deeper understanding of moment matching network. In: Proceedings of the 31st Conference on Neural Information Processing Systems (2017)
49.
Li, Y., Swersky, K., Zemel, R.: Generative moment matching networks. In: Proceedings of the 32nd International Conference on Machine Learning (2015)
50.
Liu, Q., Lee, J., Jordan, M.: A kernelized Stein discrepancy for goodness-of-fit tests. In: Proceedings of the 33rd International Conference on Machine Learning (2016)
51.
Lyne, A.M., Girolami, M., Atchadé, Y., Strathmann, H., Simpson, D.: On Russian roulette estimates for Bayesian inference with doubly-intractable likelihoods. Stat. Sci. 30(4), 443–467 (2015)MathSciNetCrossRef
52.
Matsubara, T., Knoblauch, J., Briol, F.X., Oates, C.J.: Robust generalised bayesian inference for intractable likelihoods. J. R. Stat. Soc. Ser. B 84(3), 997–1022 (2022)MathSciNetCrossRef
53.
Matsubara, T., Knoblauch, J., Briol, F.X., Oates, C.J.: Robust generalised Bayesian inference for intractable likelihoods. J. R. Stat. Soc.: Ser. B 84(3), 997–1022 (2022)MathSciNetCrossRef
54.
Mitrovic, J., Sejdinovic, D., Teh, Y.W.: DR-ABC: Approximate Bayesian computation with kernel-based distribution regression. In: Proceedings of the 33rd International Conference on Machine Learning (2016)
55.
Mroueh, Y., Li, C.L., Sercu, T., Raj, A., Cheng, Y.: Sobolev GAN. In: Proceedings of the 6th International Conference on Learning Representations (2018)
56.
Mroueh, Y., Sercu, T.: Fisher GAN. In: Proceedings of the 31st Conference on Neural Information Processing Systems (2017)
57.
Mroueh, Y., Sercu, T., Goel, V.: McGAN: mean and covariance feature matching GAN. In: Proceedings of the 34th International Conference on Machine Learning (2017)
58.
Muandet, K., Fukumizu, K., Sriperumbudur, B., Schölkopf, B.: Kernel mean embedding of distributions: A review and beyond. Found. Trends® Mach. Learn. 10(1–2), 1–141 (2017)
59.
Müller, A.: Integral probability metrics and their generating classes of functions. Adv. Appl. Probab. 29(2), 429–443 (1997)MathSciNetCrossRef
60.
Nietert, S., Goldfeld, Z., Kato, K.: Smooth \( p \)-wasserstein distance: structure, empirical approximation, and statistical applications. In: Proceedings of the 38th International Conference on Machine Learning (2021)
61.
Niu, Z., Meier, J., Briol, F.X.: Discrepancy-based inference for intractable generative models using quasi-Monte Carlo. Electron. J. Stat. 17(1), 1411–1456 (2023)MathSciNetCrossRef
62.
Oates, C.J., Girolami, M., Chopin, N.: Control functionals for Monte Carlo integration. J. R. Stat. Soc. Ser. B 79, 695–718 (2017)MathSciNetCrossRef
63.
Pardo, L.: Statistical Inference Based on Divergence Measures. Chapman and Hall/CRC (2018)
64.
Park, M., Jitkrittum, W., Sejdinovic, D.: K2-ABC: Approximate Bayesian computation with kernel embeddings. In: Proceedings of the 18th International Conference on Artificial Intelligence and Statistics (2016)
65.
Schwabik, S., Ye, G.: Topics in Banach Space Integration. World Scientific (2005)
66.
Serfling, R.J.: Approximation Theorems of Mathematical Statistics. Wiley (2009)
67.
Simon-Gabriel, C.J., Barp, A., Mackey, L.: Metrizing weak convergence with maximum mean discrepancies. J. Mach. Learn. Res. 24, 1–20 (2023)MathSciNet
68.
Sloan, I.H., Kachoyan, P.J.: Lattice methods for multiple integration: theory, error analysis and examples. SIAM J. Numer. Anal. 24(1), 116–128 (1987)MathSciNetCrossRef
69.
Sloan, I.H., Woźniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Compl. 14(1), 1–33 (1998)MathSciNetCrossRef
70.
Song, L., Zhang, X., Smola, A., Gretton, A., Schölkopf, B.: Tailoring density estimation via reproducing kernel moment matching. In: Proceedings of the 25th International Conference on Machine Learning (2008)
71.
72.
Steinwart, I., Christmann, A.: Support Vector Machines. Springer Science & Business Media (2008)
73.
Sutherland, D.J., Tung, H.Y., Strathmann, H., De, S., Ramdas, A., Smola, A.J., Gretton, A.: Generative models and model criticism via optimized maximum mean discrepancy. In: Proceedings of the 5th International Conference on Learning Representations (2017)
74.
Teymur, O., Gorham, J., Riabiz, M., Oates, C.J.: Optimal quantisation of probability measures using maximum mean discrepancy. In: Proceedings of the 24th International Conference on Artificial Intelligence and Statistics (2021)
75.
Theis, L., van den Oord, A., Bethge, M.: A note on the evaluation of generative models. In: Proceedings of the 4th International Conference on Learning Representations (2016)
76.
Van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press (2000)
77.
Wynne, G., Duncan, A.B.: A kernel two-sample test for functional data. J. Mach. Learn. Res. 23(73), 1–51 (2022)MathSciNet
78.
Wynne, G., Kasprzak, M., Duncan, A.B.: A spectral representation of kernel Stein discrepancy with application to goodness-of-fit tests for measures on infinite dimensional Hilbert spaces (2022). arXiv:​2206.​04552
Metadata
Title
Minimum Kernel Discrepancy Estimators
Author
Chris J. Oates
Copyright Year
2024
Book
Monte Carlo and Quasi-Monte Carlo Methods

Print ISBN: 978-3-031-59761-9

Electronic ISBN: 978-3-031-59762-6

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-3-031-59762-6_6

Premium Partner

    Image Credits