14,849 research outputs found
Medians of discrete sets according to a linear distance
l'URL de l'article publié est http://www.springerlink.com/link.asp?id=9rukhuabxp8abkweIn this paper, we present some results concerning the median points of a discrete set according to a distance defined by means of two directions p and q. We describe a local characterization of the median points and show how these points can be determined from the projections of the discrete set along directions p and q. We prove that the discrete sets having some connectivity properties have at most four median points according to a linear distance, and if there are four median points they form a parallelogram. Finally, we show that the 4-connected sets which are convex along the diagonal directions contain their median points along these directions
Consensus theories: an oriented survey
This article surveys seven directions of consensus theories: Arrowian results, federation consensus rules, metric consensus rules, tournament solutions, restricted domains, abstract consensus theories, algorithmic and complexity issues. This survey is oriented in the sense that it is mainly – but not exclusively – concentrated on the most significant results obtained, sometimes with other searchers, by a team of French searchers who are or were full or associate members of the Centre d'Analyse et de Mathématique Sociale (CAMS).Consensus theories ; Arrowian results ; aggregation rules ; metric consensus rules ; median ; tournament solutions ; restricted domains ; lower valuations ; median semilattice ; complexity
Computing medians and means in Hadamard spaces
The geometric median as well as the Frechet mean of points in an Hadamard
space are important in both theory and applications. Surprisingly, no
algorithms for their computation are hitherto known. To address this issue, we
use a split version of the proximal point algorithm for minimizing a sum of
convex functions and prove that this algorithm produces a sequence converging
to a minimizer of the objective function, which extends a recent result of D.
Bertsekas (2001) into Hadamard spaces. The method is quite robust and not only
does it yield algorithms for the median and the mean, but it also applies to
various other optimization problems. We moreover show that another algorithm
for computing the Frechet mean can be derived from the law of large numbers due
to K.-T. Sturm (2002). In applications, computing medians and means is probably
most needed in tree space, which is an instance of an Hadamard space, invented
by Billera, Holmes, and Vogtmann (2001) as a tool for averaging phylogenetic
trees. It turns out, however, that it can be also used to model numerous other
tree-like structures. Since there now exists a polynomial-time algorithm for
computing geodesics in tree space due to M. Owen and S. Provan (2011), we
obtain efficient algorithms for computing medians and means, which can be
directly used in practice.Comment: Corrected version. Accepted in SIAM Journal on Optimizatio
Asymptotic equivalence and adaptive estimation for robust nonparametric regression
Asymptotic equivalence theory developed in the literature so far are only for
bounded loss functions. This limits the potential applications of the theory
because many commonly used loss functions in statistical inference are
unbounded. In this paper we develop asymptotic equivalence results for robust
nonparametric regression with unbounded loss functions. The results imply that
all the Gaussian nonparametric regression procedures can be robustified in a
unified way. A key step in our equivalence argument is to bin the data and then
take the median of each bin. The asymptotic equivalence results have
significant practical implications. To illustrate the general principles of the
equivalence argument we consider two important nonparametric inference
problems: robust estimation of the regression function and the estimation of a
quadratic functional. In both cases easily implementable procedures are
constructed and are shown to enjoy simultaneously a high degree of robustness
and adaptivity. Other problems such as construction of confidence sets and
nonparametric hypothesis testing can be handled in a similar fashion.Comment: Published in at http://dx.doi.org/10.1214/08-AOS681 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Fast Hierarchical Clustering and Other Applications of Dynamic Closest Pairs
We develop data structures for dynamic closest pair problems with arbitrary
distance functions, that do not necessarily come from any geometric structure
on the objects. Based on a technique previously used by the author for
Euclidean closest pairs, we show how to insert and delete objects from an
n-object set, maintaining the closest pair, in O(n log^2 n) time per update and
O(n) space. With quadratic space, we can instead use a quadtree-like structure
to achieve an optimal time bound, O(n) per update. We apply these data
structures to hierarchical clustering, greedy matching, and TSP heuristics, and
discuss other potential applications in machine learning, Groebner bases, and
local improvement algorithms for partition and placement problems. Experiments
show our new methods to be faster in practice than previously used heuristics.Comment: 20 pages, 9 figures. A preliminary version of this paper appeared at
the 9th ACM-SIAM Symp. on Discrete Algorithms, San Francisco, 1998, pp.
619-628. For source code and experimental results, see
http://www.ics.uci.edu/~eppstein/projects/pairs
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