Open Access
2007 Interpolation of Random Hyperplanes
Ery Arias-Castro
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Electron. J. Probab. 12: 1052-1071 (2007). DOI: 10.1214/EJP.v12-435


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.


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Ery Arias-Castro. "Interpolation of Random Hyperplanes." Electron. J. Probab. 12 1052 - 1071, 2007.


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

zbMATH: 1127.60008
MathSciNet: MR2336599
Digital Object Identifier: 10.1214/EJP.v12-435

Primary: 60D05
Secondary: 62G10

Keywords: Grassmann manifold , Haar measure , Kolmogorov Entropy , pattern recognition

Vol.12 • 2007
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