Smooth Cyclically Monotone Interpolation and Empirical Center-Outward Distribution Functions
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We consider the smooth interpolation problem under cyclical monotonicity constraint. More precisely, consider finite n-tuples X={x_1,...,x_n} and Y={y_1,...,y_n} of points in R^d, and assume the existence of a unique bijection T:X→Y such that {(x,T(x)): x∈X} is cyclically monotone: our goal is to define continuous, cyclically monotone maps T̅:R^d→R^d such that T̅(x_i)=y_i, i=1,...,n, extending a classical result by Rockafellar on the subdifferentials of convex functions. Our solutions T̅ are Lipschitz, and we provide a sharp lower bound for the corresponding Lipschitz constants. The problem is motivated by, and the solution naturally applies to, the concept of empirical center-outward distribution function in R^d developed in Hallin (2018). Those empirical distribution functions indeed are defined at the observations only. Our interpolation provides a smooth extension, as well as a multivariate, outward-continuous, jump function version thereof (the latter naturally generalizes the traditional left-continuous univariate concept); both satisfy a Glivenko-Cantelli property as n→∞.
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