13,613 research outputs found
Learning the dependence structure of rare events: a non-asymptotic study
Assessing the probability of occurrence of extreme events is a crucial issue
in various fields like finance, insurance, telecommunication or environmental
sciences. In a multivariate framework, the tail dependence is characterized by
the so-called stable tail dependence function (STDF). Learning this structure
is the keystone of multivariate extremes. Although extensive studies have
proved consistency and asymptotic normality for the empirical version of the
STDF, non-asymptotic bounds are still missing. The main purpose of this paper
is to fill this gap. Taking advantage of adapted VC-type concentration
inequalities, upper bounds are derived with expected rate of convergence in
O(k^-1/2). The concentration tools involved in this analysis rely on a more
general study of maximal deviations in low probability regions, and thus
directly apply to the classification of extreme data
On Anomaly Ranking and Excess-Mass Curves
Learning how to rank multivariate unlabeled observations depending on their
degree of abnormality/novelty is a crucial problem in a wide range of
applications. In practice, it generally consists in building a real valued
"scoring" function on the feature space so as to quantify to which extent
observations should be considered as abnormal. In the 1-d situation,
measurements are generally considered as "abnormal" when they are remote from
central measures such as the mean or the median. Anomaly detection then relies
on tail analysis of the variable of interest. Extensions to the multivariate
setting are far from straightforward and it is precisely the main purpose of
this paper to introduce a novel and convenient (functional) criterion for
measuring the performance of a scoring function regarding the anomaly ranking
task, referred to as the Excess-Mass curve (EM curve). In addition, an adaptive
algorithm for building a scoring function based on unlabeled data X1 , . . . ,
Xn with a nearly optimal EM is proposed and is analyzed from a statistical
perspective
The biHecke monoid of a finite Coxeter group
The usual combinatorial model for the 0-Hecke algebra of the symmetric group
is to consider the algebra (or monoid) generated by the bubble sort operators.
This construction generalizes to any finite Coxeter group W. The authors
previously introduced the Hecke group algebra, constructed as the algebra
generated simultaneously by the bubble sort and antisort operators, and
described its representation theory.
In this paper, we consider instead the monoid generated by these operators.
We prove that it has |W| simple and projective modules. In order to construct a
combinatorial model for the simple modules, we introduce for each w in W a
combinatorial module whose support is the interval [1,w] in right weak order.
This module yields an algebra, whose representation theory generalizes that of
the Hecke group algebra. This involves the introduction of a w-analogue of the
combinatorics of descents of W and a generalization to finite Coxeter groups of
blocks of permutation matrices.Comment: 12 pages, 1 figure, submitted to FPSAC'1
Spectral gap for random-to-random shuffling on linear extensions
In this paper, we propose a new Markov chain which generalizes
random-to-random shuffling on permutations to random-to-random shuffling on
linear extensions of a finite poset of size . We conjecture that the second
largest eigenvalue of the transition matrix is bounded above by
with equality when the poset is disconnected. This Markov
chain provides a way to sample the linear extensions of the poset with a
relaxation time bounded above by and a mixing time of . We conjecture that the mixing time is in fact as for the
usual random-to-random shuffling.Comment: 16 pages, 10 figures; v2: typos fixed plus extra information in
figures; v3: added explicit conjecture 2.2 + Section 3.6 on the diameter of
the Markov Chain as evidence + misc minor improvements; v4: fixed
bibliograph
The importance of biomass net uptake for a trace metal budget in a forest stand in north-eastern France
The trace metal (TM: Cd, Cu, Ni, Pb and Zn) budget (stocks and annual fluxes) was evaluated in a forest stand (silver fir, Abies alba Miller) in north-eastern France. Trace metal concentrations were measured in different tree compartments in order to assess TM partitioning and dynamics in the trees. Inputs included bulk deposition, estimated dry deposition and weathering. Outputs were leaching and biomass exportation. Atmospheric deposition was the main input flux. The estimated dry deposition accounted for about 40% of the total trace metal deposition. The relative importance of leaching (estimated by a lumped parameter water balance model, BILJOU) and net biomass uptake (harvesting) for ecosystem exportation depended on the element. Trace metal distribution between tree compartments (stem wood and bark, branches and needles) indicated that Pb was mainly stored in the stem, whereas Zn and Ni, and to a lesser extent Cd and Cu, were translocated to aerial parts of the trees and cycled in the ecosystem. For Zn and Ni, leaching was the main output flux (N95% of the total output) and the plot budget (input–output) was negative, whereas for Pb the biomass net exportation represented 60% of the outputs and the budget was balanced. Cadmium and Cu had intermediate behaviours, with 18% and 30% of the total output relative to biomass exportation, respectively, and the budgets were negative. The net uptake by biomass was particularly important for Pb budgets, less so for Cd and Cu and not very important for Zn and Ni in such forest stands
On the representation theory of finite J-trivial monoids
In 1979, Norton showed that the representation theory of the 0-Hecke algebra
admits a rich combinatorial description. Her constructions rely heavily on some
triangularity property of the product, but do not use explicitly that the
0-Hecke algebra is a monoid algebra.
The thesis of this paper is that considering the general setting of monoids
admitting such a triangularity, namely J-trivial monoids, sheds further light
on the topic. This is a step to use representation theory to automatically
extract combinatorial structures from (monoid) algebras, often in the form of
posets and lattices, both from a theoretical and computational point of view,
and with an implementation in Sage.
Motivated by ongoing work on related monoids associated to Coxeter systems,
and building on well-known results in the semi-group community (such as the
description of the simple modules or the radical), we describe how most of the
data associated to the representation theory (Cartan matrix, quiver) of the
algebra of any J-trivial monoid M can be expressed combinatorially by counting
appropriate elements in M itself. As a consequence, this data does not depend
on the ground field and can be calculated in O(n^2), if not O(nm), where n=|M|
and m is the number of generators. Along the way, we construct a triangular
decomposition of the identity into orthogonal idempotents, using the usual
M\"obius inversion formula in the semi-simple quotient (a lattice), followed by
an algorithmic lifting step.
Applying our results to the 0-Hecke algebra (in all finite types), we recover
previously known results and additionally provide an explicit labeling of the
edges of the quiver. We further explore special classes of J-trivial monoids,
and in particular monoids of order preserving regressive functions on a poset,
generalizing known results on the monoids of nondecreasing parking functions.Comment: 41 pages; 4 figures; added Section 3.7.4 in version 2; incorporated
comments by referee in version
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