2,462 research outputs found
The arity gap of order-preserving functions and extensions of pseudo-Boolean functions
The aim of this paper is to classify order-preserving functions according to
their arity gap. Noteworthy examples of order-preserving functions are
so-called aggregation functions. We first explicitly classify the Lov\'asz
extensions of pseudo-Boolean functions according to their arity gap. Then we
consider the class of order-preserving functions between partially ordered
sets, and establish a similar explicit classification for this function class.Comment: 11 pages, material reorganize
Learning pseudo-Boolean k-DNF and Submodular Functions
We prove that any submodular function f: {0,1}^n -> {0,1,...,k} can be
represented as a pseudo-Boolean 2k-DNF formula. Pseudo-Boolean DNFs are a
natural generalization of DNF representation for functions with integer range.
Each term in such a formula has an associated integral constant. We show that
an analog of Hastad's switching lemma holds for pseudo-Boolean k-DNFs if all
constants associated with the terms of the formula are bounded.
This allows us to generalize Mansour's PAC-learning algorithm for k-DNFs to
pseudo-Boolean k-DNFs, and hence gives a PAC-learning algorithm with membership
queries under the uniform distribution for submodular functions of the form
f:{0,1}^n -> {0,1,...,k}. Our algorithm runs in time polynomial in n, k^{O(k
\log k / \epsilon)}, 1/\epsilon and log(1/\delta) and works even in the
agnostic setting. The line of previous work on learning submodular functions
[Balcan, Harvey (STOC '11), Gupta, Hardt, Roth, Ullman (STOC '11), Cheraghchi,
Klivans, Kothari, Lee (SODA '12)] implies only n^{O(k)} query complexity for
learning submodular functions in this setting, for fixed epsilon and delta.
Our learning algorithm implies a property tester for submodularity of
functions f:{0,1}^n -> {0, ..., k} with query complexity polynomial in n for
k=O((\log n/ \loglog n)^{1/2}) and constant proximity parameter \epsilon
On the Extension of Pseudo-Boolean Functions for the Aggregation of Interacting Criteria
The paper presents an analysis on the use of integrals defined for non-additive measures (or capacities) as the Choquet and the \Sipos{} integral, and the multilinear model, all seen as extensions of pseudo-Boolean functions, and used as a means to model interaction between criteria in a multicriteria decision making problem. The emphasis is put on the use, besides classical comparative information, of information about difference of attractiveness between acts, and on the existence, for each point of view, of a ``neutral level'', allowing to introduce the absolute notion of attractive or repulsive act. It is shown that in this case, the Sipos integral is a suitable solution, although not unique. Properties of the Sipos integral as a new way of aggregating criteria are shown, with emphasis on the interaction among criteria.
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
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