Electronic Journal of Probability

Interpolation of Random Hyperplanes

Ery Arias-Castro

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Grassmann Manifold Haar Measure Pattern Recognition Kolmogorov Entropy

This work is licensed under aCreative Commons Attribution 3.0 License.


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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