Asymptotic theory of greedy approximations to minimal k-pointrandom graphs
Hero, A.O., III.; Michel, O.J.J.
Information Theory, IEEE Transactions on
Volume 45, Issue 6, Sep 1999 Page(s):1921 - 1938
Digital Object Identifier 10.1109/18.782114
Summary:Let χn=(x1,...,xn), be an
independent and identically distributed (i.i.d.) sample having
multivariate distribution P. We derive almost sure (a.s.) limits for the
power-weighted edge weight function of greedy approximations to a class
of minimal graphs spanning k of the n samples. The class includes
minimal k-point graphs constructed by the partitioning method of Ravi,
Sundaram, Marathe, Rosenkrantz, and Ravi (see Proc. 5th Annu. ACM-SIAM
Symp. Discrete Algorithms, Arlington, VA, p.546-55, 1994), where the
edge weight function satisfies the quasi-additive property of Redmond
and Yukich (see Ann. Appl. Probab., vol.4, no.4, p.1057-73, 1994). In
particular, this includes greedy approximations to the k-point minimal
spanning tree (k-MST), Steiner tree (k-ST), and the traveling salesman
problem (k-TSP). An expression for the influence function of the
minimal-weight function is given which characterizes the asymptotic
sensitivity of the graph weight to perturbations in the underlying
distribution. The influence function takes a form which indicates that
the k-point minimal graph in d>1 dimensions has robustness properties
in Rd which are analogous to those of rank-order
statistics in one dimension. A direct result of our theory is that the
log-weight of the k-point minimal graph is a consistent nonparametric
estimate of the Renyi entropy of the distribution P. Possible
applications of this work include: analysis of random communication
network topologies, estimation of the mixing coefficient in
ε-contaminated mixture models, outlier discrimination and
rejection, clustering, and pattern recognition, robust nonparametric
regression, two-sample matching, and image registration
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