Optimal Rates of Convergence for Convex Set Estimation from Support Functions by Adityanand Guntuboyina
نویسنده
چکیده
The function hK is called the support function of K because it provides information on support hyperplanes and halfspaces of K . Indeed, every support halfspace of K is of the form {x :x · u≤ hK(u)} for some u ∈ Sd−1 and since K equals the intersection of all its support halfspaces, the function hK uniquely determines K . For a proof of this and other elementary properties of the support function, see Schneider [26], Section 1.7, or Rockafellar [25], Section 13. We consider the problem of estimating an unknown compact, convex set K from observations (u1, Y1), . . . , (un,Yn) drawn according to the model Yi = hK(ui)+ ξi for i = 1, . . . , n, (1)
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