Skip to main content
SpringerLink
Log in

Fitting Tractable Convex Sets to Support Function Evaluations

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

The geometric problem of estimating an unknown compact convex set from evaluations of its support function arises in a range of scientific and engineering applications. Traditional approaches typically rely on estimators that minimize the error over all possible compact convex sets; in particular, these methods allow for limited incorporation of prior structural information about the underlying set and the resulting estimates become increasingly more complicated to describe as the number of measurements available grows. We address both of these shortcomings by describing a framework for estimating tractably specified convex sets from support function evaluations. Building on the literature in convex optimization, our approach is based on estimators that minimize the error over structured families of convex sets that are specified as linear images of concisely described sets—such as the simplex or the spectraplex—in a higher-dimensional space that is not much larger than the ambient space. Convex sets parametrized in this manner are significant from a computational perspective as one can optimize linear functionals over such sets efficiently; they serve a different purpose in the inferential context of the present paper, namely, that of incorporating regularization in the reconstruction while still offering considerable expressive power. We provide a geometric characterization of the asymptotic behavior of our estimators, and our analysis relies on the property that certain sets which admit semialgebraic descriptions are Vapnik–Chervonenkis classes. Our numerical experiments highlight the utility of our framework over previous approaches in settings in which the measurements available are noisy or small in number as well as those in which the underlying set to be reconstructed is non-polyhedral.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Notes

  1. We note that this is the case even though the estimator \({\hat{K}}^{\mathrm {LSE}}_n\) is consistent; in particular, consistency simply refers to the convergence as \(n \rightarrow \infty \) of \({\hat{K}}^{\mathrm {LSE}}_n\) to \(K^\star \) in a topological sense (e.g., in Hausdorff distance) and it does not provide any information about the facial structure of \({\hat{K}}^{\mathrm {LSE}}_n\) relative to that of \(K^\star \).

  2. The Tammes problem is a special case of Thompson’s problem as well as Smale’s 7th problem [29].

References

  1. Averkov, G.: Optimal size of linear matrix inequalities in semidefinite approaches to polynomial optimization. SIAM J. Appl. Algebra Geom. 3(1), 128–151 (2019)

    Article  MathSciNet  Google Scholar 

  2. Balázs, G.: Convex Regression: Theory, Practice, and Applications. PhD thesis, University of Alberta (2016)

  3. Barvinok, A.: Approximations of convex bodies by polytopes and by projections of spectrahedra (2012). arXiv:1204.0471

  4. Ben-Tal, A., Nemirovski, A.: On polyhedral approximations of the second-order cone. Math. Oper. Res. 26(2), 193–205 (2001)

    Article  MathSciNet  Google Scholar 

  5. Bronstein, E.M.: Approximation of convex sets by polytopes. J. Math. Sci. (N.Y.) 153(6), 727–762 (2008)

    Article  MathSciNet  Google Scholar 

  6. Cai, T.T., Guntuboyina, A., Wei, Y.: Adaptive estimation of planar convex sets. Ann. Stat. 46(3), 1018–1049 (2018)

    Article  MathSciNet  Google Scholar 

  7. Danzer, L.: Finite point-sets on \(S^2\) with minimum distance as large as possible. Discrete Math. 60, 3–66 (1986)

    Article  MathSciNet  Google Scholar 

  8. Fisher, N.I., Hall, P., Turlach, B.A., Watson, G.S.: On the estimation of a convex set from noisy data on its support function. J. Am. Stat. Assoc. 92(437), 84–91 (1997)

    Article  MathSciNet  Google Scholar 

  9. Gaillard, F.: Normal chest CT-lung window. Radiopaedia. https://radiopaedia.org/cases/normal-chest-ct-lung-window

  10. Gardner, R.J., Kiderlen, M.: A new algorithm for 3D reconstruction from support functions. IEEE Trans. Pattern Anal. Mach. Intell. 31(3), 556–562 (2009)

    Article  MathSciNet  Google Scholar 

  11. Gardner, R.J., Kiderlen, M., Milanfar, P.: Convergence of algorithms for reconstructing convex bodies and directional measures. Ann. Stat. 34(3), 1331–1374 (2006)

    Article  MathSciNet  Google Scholar 

  12. Gouveia, J., Parrilo, P.A., Thomas, R.R.: Lifts of convex sets and cone factorizations. Math. Oper. Res. 38(2), 248–264 (2013)

    Article  MathSciNet  Google Scholar 

  13. Gregor, J., Rannou, F.R.: Three-dimensional support function estimation and application for projection magnetic resonance imaging. Int. J. Imaging Syst. Technol. 12(1), 43–50 (2002)

    Article  Google Scholar 

  14. Guntuboyina, A.: Optimal rates of convergence for convex set estimation from support functions. Ann. Stat. 40(1), 385–411 (2012)

    Article  MathSciNet  Google Scholar 

  15. Hall, P., Turlach, B.A.: On the estimation of a convex set with corners. IEEE Trans. Pattern Anal. Mach. Intell. 21(3), 225–234 (1999)

    Article  Google Scholar 

  16. Hannah, L.A., Dunson, D.B.: Multivariate convex regression with adaptive partitioning. J. Mach. Learn. Res. 14, 3261–3294 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Kosorok, M.R.: Introduction to Empirical Processes and Semiparametric Inference. Springer Series in Statistics. Springer, New York (2008)

    MATH  Google Scholar 

  18. Lele, A.S., Kulkarni, S.R., Willsky, A.S.: Convex-polygon estimation from support-line measurements and applications to target reconstruction from laser-radar data. J. Optical Soc. Amer. A 9(10), 1693–1714 (1992)

    Article  Google Scholar 

  19. Lloyd, S.P.: Least squares quantization in PCM. IEEE Trans. Inf. Theory 28(2), 129–137 (1982)

    Article  MathSciNet  Google Scholar 

  20. Magnani, A., Boyd, S.P.: Convex piecewise-linear fitting. Optim. Eng. 10(1), 1–17 (2009)

    Article  MathSciNet  Google Scholar 

  21. Nesterov, Yu., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics, vol. 13. Society for Industrial and Applied Mathematics, Philadelphia (1994)

  22. Pollard, D.: Strong consistency of \(k\)-means clustering. Ann. Stat. 9(1), 135–140 (1981)

    Article  MathSciNet  Google Scholar 

  23. Pollard, D.: Convergence of Stochastic Processes. Springer Series in Statistics. Springer, New York (1984)

    Google Scholar 

  24. Prince, J.L., Willsky, A.S.: Reconstructing convex sets from support line measurements. IEEE Trans. Pattern Anal. Mach. Intell. 12(4), 377–389 (1990)

    Article  Google Scholar 

  25. Prince, J.L., Willsky, A.S.: Convex set reconstruction using prior shape information. CVGIP: Graph. Models Image Process. 53(5), 413–427 (1991)

    Google Scholar 

  26. Saunderson, J.: Limitations on the expressive power of convex cones without long chains of faces. SIAM J. Optim. 30(1), 1033–1047 (2020)

    Article  MathSciNet  Google Scholar 

  27. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 151. Cambridge University Press, Cambridge (2014)

  28. Schütte, K., van der Waerden, B.L.: Auf welcher Kugel haben \(5\), \(6\), \(7\), \(8\) oder \(9\) Punkte mit Mindestabstand Eins Platz? Math. Ann. 123, 96–124 (1951)

    Article  MathSciNet  Google Scholar 

  29. Smale, S.: Mathematical problems for the next century. Math. Intell. 20(2), 7–15 (1998)

    Article  MathSciNet  Google Scholar 

  30. Stark, H., Peng, H.: Shape estimation in computer tomography from minimal data. J. Opt. Soc. Am. A 5(3), 331–343 (1988)

    Article  Google Scholar 

  31. Stengle, G., Yukich, J.E.: Some new Vapnik–Chervonenkis classes. Ann. Stat. 17(4), 1441–1446 (1989)

    Article  MathSciNet  Google Scholar 

  32. Tammes, P.M.L.: On the Origin of Number and Arrangement of the Places of Exit on the Surface of Pollen-Grains. Recueil des Travaux Botaniques Néerlandais, vol. 27. Koninklijke Nederlandse Botanische Vereniging (1930)

  33. Vapnik, V.N., Chervonenkis, A.Ya.: On the uniform convergence of relative frequencies of events to their probabilities. Proc. USSR Acad. Sci. 181(4), 781–783 (1968). (in Russian)

  34. Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of High Performance Computing, 1 Fusionopolis Way, # 16-16 Connexis, Singapore, 138632, Singapore

    Yong Sheng Soh

  2. Department of Computing and Mathematical Sciences and Department of Electrical Engineering, California Institute of Technology, Pasadena, CA, 91125, USA

    Venkat Chandrasekaran

Authors
  1. Yong Sheng Soh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Venkat Chandrasekaran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Sheng Soh.

Additional information

Editor in Charge: Kenneth Clarkson

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors were supported in part by NSF grants CCF-1350590 and CCF-1637598, by Air Force Office of Scientific Research Grant FA9550-16-1-0210, by a Sloan research fellowship, and an A*STAR (Agency for Science, Technology and Research, Singapore) fellowship.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Soh, Y.S., Chandrasekaran, V. Fitting Tractable Convex Sets to Support Function Evaluations. Discrete Comput Geom 66, 510–551 (2021). https://doi.org/10.1007/s00454-020-00258-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-020-00258-0

Keywords

Search

Navigation