Skip to main content
SpringerLink
Log in

Variational Bayesian data analysis on manifold

  • Published:
Control Theory and Technology Aims and scope Submit manuscript

Abstract

In this paper, variational inference is studied on manifolds with certain metrics. To solve the problem, the analysis is first proposed for the variational Bayesian on Lie group, and then extended to the manifold that is approximated by Lie groups. Then the convergence of the proposed algorithm with respect to the manifold metric is proved in two iterative processes: variational Bayesian expectation (VB-E) step and variational Bayesian maximum (VB-M) step. Moreover, the effective of different metrics for Bayesian analysis is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Bishop. Pattern Recognition and Machine Learning. New York: Springer, 2006.

    MATH  Google Scholar 

  2. M. Wainwright, M. Jordan. Graphical models, exponential families, and variational inference. Foundations and Trends in Machine Learning, 2008, 1(1/2): 1–305.

    MATH  Google Scholar 

  3. M. J. Beal. Variational Algorithms for Approximate Bayesian Inference. London: University of London, 2003.

    Google Scholar 

  4. G. Casella, R. L. Berger. Statistical Inference. Grove: Duxbury Pacific, 2002.

    MATH  Google Scholar 

  5. H. Lan, X. Wang, Q. Pan, et al. A survey on joint tracking using expectation-maximization based techniques. Information Fusion, 2016, 30(1): 52–68.

    Article  Google Scholar 

  6. M. A. Sato. Online model selection based on the variational bayes. Neural Computation, 2001, 13(7): 1649–1681.

    Article  MATH  Google Scholar 

  7. S. Sarkka, A. Nummenmaa. Recursive noise adaptive kalman filtering by variational bayesian approximations. IEEE Transactions on Automatic Control, 2009, 54(3): 596–600.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Agamennoni, J. I. Nieto, E. M. Nebot. Approximate inference in state-space models with heavy-tailed noise. IEEE Transactions on Signal Processing, 2012, 60(10): 5024–5037.

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Zhu, H. Leung, Z. He. State estimation in unknown non-gaussian measurement noise using variational bayesian technique. IEEE Transactions on Aerospace and Electronic Systems, 2013, 49(4): 2601–2614.

    Article  Google Scholar 

  10. C. Fefferman, S. Mitter, H. Narayanan. Testing the manifold hypothesis. Journal of the American Mathematical Society, 2013, 29(4): 1786–1794.

    MathSciNet  MATH  Google Scholar 

  11. G. Chirikjian, M. Kobilarov. Gaussian approximation of non-linear measurement models on Lie groups. Proceedings of the 53rd IEEE Conference on Decision and Control, Los Angeles: IEEE, 2014: 6401–6407.

    Chapter  Google Scholar 

  12. G. Chirikjian. Information-theoretic inequalities on unimodular Lie groups. Journal of Geometric Mechanics, 2017, 2(2): 119–158.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Bronstein, J. Bruna, Y. LeCun, et al. Geometric deep learning: going beyond Euclidean data. IEEE Signal Processing Magazine, 2017, 34(4): 18–42.

    Article  Google Scholar 

  14. Y. Wang, G. S. Chirikjian. Nonparametric second-order theory of error propagation on the Euclidean group. International Journal of Robotics Research, 2008, 27(11): 1258–1273.

    Article  Google Scholar 

  15. A. S. Willsky. Some estimation problems on Lie groups. Geometric Methods in System Theory. D. Q. Mayne and R. W. Brockett (eds.). Dordrecht: Springer, 1973: 305–314.

    Chapter  Google Scholar 

  16. D. Bump. Lie Groups. 2nd ed. New York: Springer, 2013.

    Book  MATH  Google Scholar 

  17. L. Koralov, Y. Sinai. Theory of Probability and Random Processes. Berlin: Springer, 2007.

    Book  MATH  Google Scholar 

  18. A. Gelman, J. B. Carlin, H. S. Stern, et al. Bayesian Data Analysis, Boca Raton: CRC Press, 2014.

    MATH  Google Scholar 

  19. W. Rudin. Fourier Analysis on Groups. Dover Books on Mathematics. New York: Dover Publications, 2017.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Lab of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yang Ming

  2. University of Chinese Academy of Sciences, Beijing, 100190, China

    Yang Ming

Authors
  1. Yang Ming
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yang Ming.

Additional information

This work was supported by the National Key Research and Development Program of China (No. 2016YF-B0901900) and the National Natural Science Foundation of China (Nos. 61733018, 61333001, 61573344).

Yang Ming received his B.Sc. degree from Northwest University in 2015. He is currently a Ph.D. candidate in Academy of Mathematics and Systems Science, Chinese Academy of Sciences. His research interests include machine learning and number theory.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ming, Y. Variational Bayesian data analysis on manifold. Control Theory Technol. 16, 212–220 (2018). https://doi.org/10.1007/s11768-018-8083-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11768-018-8083-4

Keywords

Search

Navigation