2,612 research outputs found
Measuring the interactions among variables of functions over the unit hypercube
By considering a least squares approximation of a given square integrable
function by a multilinear polynomial of a specified
degree, we define an index which measures the overall interaction among
variables of . 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. We show that this interaction index has
appealing properties which naturally generalize the properties of the Banzhaf
interaction index. In particular, we interpret this index as an expected value
of the difference quotients of or, under certain natural conditions on ,
as an expected value of the derivatives of . These interpretations show a
strong analogy between the introduced interaction index and the overall
importance index defined by Grabisch and Labreuche [7]. Finally, we discuss a
few applications of the interaction index
Approximations of Lovasz extensions and their induced interaction index
The Lovasz extension of a pseudo-Boolean function is
defined on each simplex of the standard triangulation of as the
unique affine function that interpolates at the
vertices of the simplex. Its degree is that of the unique multilinear
polynomial that expresses . In this paper we investigate the least squares
approximation problem of an arbitrary Lovasz extension by Lovasz
extensions of (at most) a specified degree. We derive explicit expressions of
these approximations. The corresponding approximation problem for
pseudo-Boolean functions was investigated by Hammer and Holzman (1992) and then
solved explicitly by Grabisch, Marichal, and Roubens (2000), giving rise to an
alternative definition of Banzhaf interaction index. Similarly we introduce a
new interaction index from approximations of and we present some of
its properties. It turns out that its corresponding power index identifies with
the power index introduced by Grabisch and Labreuche (2001).Comment: 19 page
Weighted Banzhaf power and interaction indexes through weighted approximations of games
The 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. We show that these indexes define a subclass of the family of
probabilistic interaction indexes and study their most important properties.
Finally, we give an interpretation of the Banzhaf and Shapley interaction
indexes as centers of mass of this subclass of interaction indexes
Limitations of semidefinite programs for separable states and entangled games
Semidefinite programs (SDPs) are a framework for exact or approximate
optimization that have widespread application in quantum information theory. We
introduce a new method for using reductions to construct integrality gaps for
SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy
in approximating two particularly important sets in quantum information theory,
where previously no -round integrality gaps were known: the set of
separable (i.e. unentangled) states, or equivalently, the
norm of a matrix, and the set of quantum correlations; i.e. conditional
probability distributions achievable with local measurements on a shared
entangled state. In both cases no-go theorems were previously known based on
computational assumptions such as the Exponential Time Hypothesis (ETH) which
asserts that 3-SAT requires exponential time to solve. Our unconditional
results achieve the same parameters as all of these previous results (for
separable states) or as some of the previous results (for quantum
correlations). In some cases we can make use of the framework of
Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not
only the SoS hierarchy. Our hardness result on separable states also yields a
dimension lower bound of approximate disentanglers, answering a question of
Watrous and Aaronson et al. These results can be viewed as limitations on the
monogamy principle, the PPT test, the ability of Tsirelson-type bounds to
restrict quantum correlations, as well as the SDP hierarchies of
Doherty-Parrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz.Comment: 47 pages. v2. small changes, fixes and clarifications. published
versio
Symmetric approximations of pseudo-Boolean functions with applications to influence indexes
We introduce an index for measuring the influence of the k-th smallest
variable on a pseudo-Boolean function. This index is defined from a weighted
least squares approximation of the function by linear combinations of order
statistic functions. We give explicit expressions for both the index and the
approximation and discuss some properties of the index. Finally, we show that
this index subsumes the concept of system signature in engineering reliability
and that of cardinality index in decision making
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Least Square Approximations and Conic Values of Cooperative Games
URL des Documents de travail : http://centredeconomiesorbonne.univ-paris1.fr/documents-de-travail/Documents de travail du Centre d'Economie de la Sorbonne 2015.47 - ISSN : 1955-611XThe problem of least square approximation for set functions by set functions satisfying specified linear equality or inequality constraints is considered. The problem has important applications in the field of pseudo-Boolean functions, decision making and in cooperative game theory, where approximation by additive set functions yields so-called least square values. In fact, it is seem that every linear value for cooperative games arises from least square approximation. We provide a general approach and problem overview. In particular, we derive explicit formulas for solutions under mild constraints, which include and extend previous results in the literature.On considère le problème de l'approximation au sens des moindres carrés des fonctions d'ensemble par des fonctions d'ensemble satisfaisant des contraintes linéaires d'égalité ou d'inégalité. Le problème a des applications importantes dans le domaine des fonctions pseudo-Booléennes, la décision et la théorie des jeux coopératifs, où l'approximation par des jeux additifs mène à la notion de valeur aux moindres carrés. En fait, on voit que toute valeur linéaire pour les jeux coopératifs vient d'un problème d'approximation par les moindres carrés. Nous proposons une approche générale du problème. En particulier, nous obtenons des formules explicites pour les solutions sous des hypothèses faibles, qui incluent et étendent des résultats précédents de la littérature
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