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Interpolation of Random Hyperplanes


 
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1. Title Title of document Interpolation of Random Hyperplanes
 
2. Creator Author's name, affiliation, country Ery Arias-Castro; University of California, San Diego
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Grassmann Manifold; Haar Measure; Pattern Recognition; Kolmogorov Entropy
 
3. Subject Subject classification 60D05; 62G10
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) NSF grant DMS-0603890 and the Mathematical Sciences Research Institute
 
7. Date (YYYY-MM-DD) 2007-08-15
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/435
 
10. Identifier Digital Object Identifier 10.1214/EJP.v12-435
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 12
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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