23,781 research outputs found
On the Number of Facets of Polytopes Representing Comparative Probability Orders
Fine and Gill (1973) introduced the geometric representation for those
comparative probability orders on n atoms that have an underlying probability
measure. In this representation every such comparative probability order is
represented by a region of a certain hyperplane arrangement. Maclagan (1999)
asked how many facets a polytope, which is the closure of such a region, might
have. We prove that the maximal number of facets is at least F_{n+1}, where F_n
is the nth Fibonacci number. We conjecture that this lower bound is sharp. Our
proof is combinatorial and makes use of the concept of flippable pairs
introduced by Maclagan. We also obtain an upper bound which is not too far from
the lower bound.Comment: 13 page
Noisy Submodular Maximization via Adaptive Sampling with Applications to Crowdsourced Image Collection Summarization
We address the problem of maximizing an unknown submodular function that can
only be accessed via noisy evaluations. Our work is motivated by the task of
summarizing content, e.g., image collections, by leveraging users' feedback in
form of clicks or ratings. For summarization tasks with the goal of maximizing
coverage and diversity, submodular set functions are a natural choice. When the
underlying submodular function is unknown, users' feedback can provide noisy
evaluations of the function that we seek to maximize. We provide a generic
algorithm -- \submM{} -- for maximizing an unknown submodular function under
cardinality constraints. This algorithm makes use of a novel exploration module
-- \blbox{} -- that proposes good elements based on adaptively sampling noisy
function evaluations. \blbox{} is able to accommodate different kinds of
observation models such as value queries and pairwise comparisons. We provide
PAC-style guarantees on the quality and sampling cost of the solution obtained
by \submM{}. We demonstrate the effectiveness of our approach in an
interactive, crowdsourced image collection summarization application.Comment: Extended version of AAAI'16 pape
Biased quantitative measurement of interval ordered homothetic preferences
We represent interval ordered homothetic preferences with a quantitative homothetic utility function and a multiplicative bias. When preferences are weakly ordered (i.e. when indifference is transitive), such a bias equals 1. When indifference is intransitive, the biasing factor is a positive function smaller than 1 and measures a threshold of indifference. We show that the bias is constant if and only if preferences are semiordered, and we identify conditions ensuring a linear utility function. We illustrate our approach with indifference sets on a two dimensional commodity space.Weak order, semiorder, interval order, intransitive indifference, independence, homothetic, representation, linear utility
Combining decision procedures for the reals
We address the general problem of determining the validity of boolean
combinations of equalities and inequalities between real-valued expressions. In
particular, we consider methods of establishing such assertions using only
restricted forms of distributivity. At the same time, we explore ways in which
"local" decision or heuristic procedures for fragments of the theory of the
reals can be amalgamated into global ones. Let Tadd[Q] be the
first-order theory of the real numbers in the language of ordered groups, with
negation, a constant 1, and function symbols for multiplication by
rational constants. Let Tmult[Q] be the analogous theory for the
multiplicative structure, and let T[Q] be the union of the two. We
show that although T[Q] is undecidable, the universal fragment of
T[Q] is decidable. We also show that terms of T[Q]can
fruitfully be put in a normal form. We prove analogous results for theories in
which Q is replaced, more generally, by suitable subfields F
of the reals. Finally, we consider practical methods of establishing
quantifier-free validities that approximate our (impractical) decidability
results.Comment: Will appear in Logical Methods in Computer Scienc
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