Interpolation of Random Hyperplanes

2006/09/12 by Arias-Castro, Ery
#60D05 #62G10 #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.math/0609340

Abstract

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

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