Interpolation of Random Hyperplanes
Abstract
Let (Zi,Wi):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 Rd. 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 H\"older type, we obtain probability bounds on the maximum number of points a function f in F interpolates.
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