Asymptotic Relations Between Minimal Graphs and α-entropy

This report is concerned with power-weighted weight functionals associated with a minimal graph spanning a random sample of n points from a general multivariate Lebesgue density f over [0, 1]. It is known that under broad conditions, when the functional applies power exponent γ ∈ (1, d) to the graph edge lengths, the log of the functional normalized by n(d−γ)/d is a strongly consistent estimator of the Rényi entropy of order α = (d−γ)/d. In this paper, we investigate almost sure (a.s.) and Lκ-norm (r.m.s. for κ = 2) convergence rates of this functional. In particular, when 1 ≤ γ ≤ d − 1, we show that over the space of compacted supported multivariate densities f such that f ∈ Σd(β, L) (the space of Holder continuous functions), 0 < β ≤ 1, the Lκ-norm convergence rate is bounded above by O ( n−αβ/(αβ+1) 1/d) ) . We obtain similar rate bounds for minimal graph approximations implemented by a progressive divide-and-conquer partitioning heuristic. We also obtain asymptotic lower bounds for the respective rates of convergence, using minimax techniques from nonparametric function estimation. In addition to Euclidean optimization problems, these results have application to non-parametric entropy and information divergence estimation; adaptive vector quantization; and pattern recognition.