Electronic Journal of Probability

Interpolation of Random Hyperplanes

Ery Arias-Castro

Full-text: Open access

Abstract

Let $\{(Z_i,W_i):i=1,\dots,n\}$ be uniformly distributed in $[0,1]^d \times \mathbb{G}(k,d)$, where $\mathbb{G}(k,d)$ denotes the space of $k$-dimensional linear subspaces of $\mathbb{R}^d$. For a differentiable function $f: [0,1]^k \rightarrow [0,1]^d$, we say that $f$ interpolates $(z,w) \in [0,1]^d \times \mathbb{G}(k,d)$ if there exists $x \in [0,1]^k$ such that $f(x) = z$ and $\vec{f}(x) = w$, where $\vec{f}(x)$ denotes the tangent space at $x$ defined by $f$. For a smoothness class ${\cal F}$ of Holder type, we obtain probability bounds on the maximum number of points a function $f \in {\cal F}$ interpolates.

Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 38, 1052-1071.

Dates
Accepted: 15 August 2007
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464818509

Digital Object Identifier
doi:10.1214/EJP.v12-435

Mathematical Reviews number (MathSciNet)
MR2336599

Zentralblatt MATH identifier
1127.60008

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 62G10: Hypothesis testing

Keywords
Grassmann Manifold Haar Measure Pattern Recognition Kolmogorov Entropy

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Arias-Castro, Ery. Interpolation of Random Hyperplanes. Electron. J. Probab. 12 (2007), paper no. 38, 1052--1071. doi:10.1214/EJP.v12-435. https://projecteuclid.org/euclid.ejp/1464818509


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References

  • P.-A. Absil, A. Edelman, and P. Koev. On the largest principal angle between random subspaces. Linear Algebra Appl., 414 (2006), no. 1, 288-294.
  • E. Arias-Castro, D. L. Donoho, X. Huo, and C. Tovey. Connect-the-dots: How many random points can a regular curve pass through? Adv. in Appl. Probab., 37 (2005), no. 3, 571-603.
  • E. Arkin, J. Mitchell, and G. Narasimhan. Resource-constrained geometric network optimization. In Proc. of ACM Symposium on Computational Geometry, Minneapolis, (1997), 307-316.
  • B. Awerbuch, Y. Azar, A. Blum, and S. Vempala. New approximation guarantees for minimum-weight k-trees and prize-collecting salesmen. SIAM J. Comput., 28 (1999), no. 1, 254-262.
  • B. DasGupta, J. Hespanha, and E. Sontag. Computational complexities of honey-pot searching with local sensory information. In 2004 American Control Conference (ACC 2004), pages 2134-2138. 2004.
  • D. Field, A. Hayes, and R. Hess. Contour integration by the human visual system: evidence for a local association field. Vision Research, 33(2):173-193, 1993.
  • G. H. Golub and C. F. Van Loan. Matrix computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, third edition, 1996.
  • X. Huo, D. Donoho, C. Tovey, and E. Arias-Castro. Dynamic programming methods for `connecting the dots' in scattered point clouds. INFORMS J. Comput., 2005. To appear.
  • A. N. Kolmogorov. Selected works of A. N. Kolmogorov. Vol. III, volume 27 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1993.
  • V. D. Milman and G. Schechtman. Asymptotic theory of finite-dimensional normed spaces, volume 1200 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov.
  • F. Mosteller. Fifty challenging problems in probability with solutions. Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965.
  • G. R. Shorack and J. A. Wellner. Empirical processes with applications to statistics. John Wiley & Sons Inc., New York, 1986.