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Interpolation of Random Hyperplanes

   
@article{EJP435,
	author = {Ery Arias-Castro},
	title = {Interpolation of Random Hyperplanes},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {12},
	year = {2007},
	keywords = {Grassmann Manifold; Haar Measure; Pattern Recognition; Kolmogorov Entropy},
	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.},
	pages = {no. 38, 1052-1071},
	issn = {1083-6489},
	doi = {10.1214/EJP.v12-435},    
        url = {http://ejp.ejpecp.org/article/view/435}}