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Revisiting the Approximate Carathéodory Problem via the Frank-Wolfe Algorithm
[article]
2021
arXiv
pre-print
The approximate Carathéodory theorem states that given a compact convex set 𝒞⊂ℝ^n and p∈[2,+∞[, each point x^*∈𝒞 can be approximated to ϵ-accuracy in the ℓ_p-norm as the convex combination of 𝒪(pD_p^2/ϵ^2) vertices of 𝒞, where D_p is the diameter of 𝒞 in the ℓ_p-norm. A solution satisfying these properties can be built using probabilistic arguments or by applying mirror descent to the dual problem. We revisit the approximate Carathéodory problem by solving the primal problem via the Frank-Wolfe
arXiv:1911.04415v5
fatcat:ymeeb5d4bjezre7cw43rn3vyuy