Pierre Mathonet
Symmetric approximations of pseudo-Boolean functions.
Fri, 04/16/2010 - 15:01 — SomNambulAuthors: Jean-Luc Marichal, Pierre Mathonet
Subjects: Optimization and Control
AbstractWe consider the approximation problem of a pseudo-Boolean function by a
symmetric pseudo-Boolean function in the sense of weighted least squares. We
give explicit expressions for the approximation and provide interpretations and
properties of its L-statistic representation. We also discuss applications of
these expressions in cooperative game theory and engineering reliability.Projectively equivariant quantizations over the superspace $\R^{p|q}$.
Thu, 03/18/2010 - 16:01 — SomNambulAuthors: Pierre Mathonet, Fabian Radoux
Subjects: Differential Geometry
AbstractIn this paper, we analyze the question of existence of a projectively
equivariant quantization and symbol maps in the framework of super projective
geometry. We show that the methods and results introduced in [18,8,1,2] in the
purely even situation can be generalized to show the existence and uniqueness
of such an equivariant quantization map, except in some so-called critical
situations. We also provide explicit formulas in terms of a generalized
divergence operator acting on supersymmetric tensor fields.Weighted Banzhaf interaction index through weighted approximations of games.
Tue, 01/19/2010 - 16:01 — SomNambulAuthors: Jean-Luc Marichal, Pierre Mathonet
Subjects: Optimization and Control
AbstractThe Banzhaf power index was introduced in cooperative game theory to measure
the real power of players in a game. The Banzhaf interaction index was then
proposed to measure the interaction degree inside coalitions of players. It was
shown that the power and interaction indexes can be obtained as solutions of a
standard least squares approximation problem for pseudo-Boolean functions.
Considering certain weighted versions of this approximation problem, we define
a class of weighted interaction indexes that generalize the Banzhaf interaction
index.- Read more
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Measuring the interactions among variables of functions over the unit hypercube.
Wed, 12/09/2009 - 16:01 — SomNambulAuthors: Jean-Luc Marichal, Pierre Mathonet
Subjects: Optimization and Control
AbstractBy considering a least squares approximation of a given square integrable
function $f\colon[0,1]^n\to\R$ by a multilinear polynomial of a specified
degree, we define an index which measures the overall interaction among
variables of $f$. This definition extends the concept of Banzhaf interaction
index introduced in cooperative game theory. Our approach is partly inspired
from multilinear regression analysis, where interactions among the independent
variables are taken into consideration.
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