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