Monotone Set Function Research Articles

Conditional aggregation operators defined by the Choquet integral and the Sugeno integral with respect to general fractal measures

Conditional aggregation operators are defined by the Choquet integral and the Sugeno integral with respect to a monotone set function that assesses positive measure of the conditioning set. General Hausdorff and packing measures are introduced and examples of infinite s-sets with positive and finite generalized Hausdorff and packing measures are constructed and their fractal dimensions are compared. Coherent upper conditional previsions on the linear space of all Choquet integrable random variables are defined by the Choquet integral with respect to the general Hausdorff and packing measures when the conditioning event has positive and finite generalized Hausdorff and packing measures in its respective fractal dimensions. Conditional aggregation operators are defined by the Sugeno integral with respect to general Hausdorff and packing measures on the class of all Sugeno integrable random variables. Actually, the general Hausdorff and packing dimensions are proven to be the Sugeno integral with respect to the Lebesgue measure of the general Hausdorff and packing measures respectively.

Improved Linear-Time Streaming Algorithms for Maximizing Monotone Cardinality-Constrained Set Functions

Many real-world applications arising from social networks, personalized recommendations, and others, require extracting a relatively small but broadly representative portion of massive data sets. Such problems can often be formulated as maximizing a monotone set function with cardinality constraints. In this paper, we consider a streaming model where elements arrive quickly over time, and create an effective, and low-memory algorithm. First, we provide the first single-pass linear-time algorithm, which is a a deterministic algorithm, achieves an approximation ratio of [Formula: see text] for any [Formula: see text] with a query complexity of [Formula: see text] and a memory complexity of [Formula: see text], where [Formula: see text] is a positive integer and [Formula: see text] is the submodularity ratio. However, the algorithm may produce less-than-ideal results. Our next result is to describe a multi-streaming algorithm, which is the first deterministic algorithm to attain an approximation ratio of [Formula: see text] with linear query complexity.

New horizon in fuzzy distributions: statistical distributions in continuous domains generated by Choquet integral.

In this paper, some statistical properties of the Choquet integral are discussed. As an interesting application of Choquet integral and fuzzy measures, we introduce a new class of exponential-like distributions related to monotone set functions, called Choquet exponential distributions, by combining the properties of Choquet integral with the exponential distribution. We show some famous statistical distributions such as gamma, logistic, exponential, Rayleigh and other distributions are a special class of Choquet distributions. Then, we show that this new proposed Choquet exponential distribution is better on daily gold price data analysis. Also, a real dataset of the daily number of new infected people to coronavirus in the USA in the period of 2020/02/29 to 2020/10/19 is analyzed. The method presented in this article opens a new horizon for future research.

Two approximation algorithms for maximizing nonnegative weakly monotonic set functions

Two approximation algorithms for maximizing nonnegative weakly monotonic set functions

Quantitative approximation by nonlinear convolution operators of Landau-Choquet type

By using the concept of Choquet nonlinear integral with respect to a monotone set function, we introduce the nonlinear convolution operators of Landau-Choquet type, with respect to a family of submodular set functions. Quantitative approximation results in terms of the modulus of continuity are obtained with respect to some particular possibility measures. For some subclasses of functions we prove that these Landau-Choquet type operators can have essentially better approximation properties than their classical correspondents.

Maximize a monotone function with a generic submodularity ratio

Generic submodularity ratio γ is a general measurement to characterize how close a nonnegative monotone set function is to be submodular. In this paper, we make a systematic analysis of greedy algorithms for maximizing a monotone and normalized set function with a generic submodularity ratio γ under Cardinality constraints, Knapsack constraints, Matroid constraints and K-intersection constraints.

Integral Representation of Coherent Lower Previsions by Super-Additive Integrals

Coherent lower previsions generalize the expected values and they are defined on the class of all real random variables on a finite non-empty set. Well known construction of coherent lower previsions by means of lower probabilities, or by means of super-modular capacities-based Choquet integrals, do not cover this important class of functionals on real random variables. In this paper, a new approach to the construction of coherent lower previsions acting on a finite space is proposed, exemplified and studied. It is based on special decomposition integrals recently introduced by Even and Lehrer, in our case the considered decomposition systems being single collections and thus called collection integrals. In special case when these integrals, defined for non-negative random variables only, are shift-invariant, we extend them to the class of all real random variables, thus obtaining so called super-additive integrals. Our proposed construction can be seen then as a normalized super-additive integral. We discuss and exemplify several particular cases, for example, when collections determine a coherent lower prevision for any monotone set function. For some particular collections, only particular set functions can be considered for our construction. Conjugated coherent upper previsions are also considered.

Randomized Greedy Sensor Selection: Leveraging Weak Submodularity

We study the problem of estimating a random process from the observations collected by a network of sensors that operate under resource constraints. When the dynamics of the process and sensor observations are described by a state-space model and the resource are unlimited, the conventional Kalman filter provides the minimum mean square error (MMSE) estimates. However, at any given time, restrictions on the available communications bandwidth and computational capabilities and/or power impose a limitation on the number of network nodes, whose observations can be used to compute the estimates. We formulate the problem of selecting the most informative subset of the sensors as a combinatorial problem of maximizing a monotone set function under a uniform matroid constraint. For the MMSE estimation criterion, we show that the maximum elementwise curvature of the objective function satisfies a certain upper-bound constraint and is, therefore, weak submodular. Building upon the work of Mirzasoleiman et al. on submodular maximization, we develop an efficient randomized greedy algorithm for sensor selection and establish guarantees on the estimator's performance in this setting. Extensive simulation results demonstrate the efficacy of the randomized greedy algorithm compared to state-of-the-art greedy and semidefinite programming relaxation methods.

Minimizing Ratio of Monotone Non-submodular Functions

In this paper, we investigate the problem of minimizing the ratio of normalized non-negative monotone non-submodular set function f and normalized non-negative monotone set function g. We take advantage of the greedy technique and get a performance guarantee depending on the generalized curvature and inverse generalized curvature of f, as well as the submodularity ratio of g. Our results generalize the works of Bai et al. (Algorithms for optimizing the ratio of submodular functions. In: Proceedings of the 33rd International Conference on Machine Learning, 2016) and Qian et al. (Optimizing ratio of monotone set functions. In: Proceedings of the 26th International Joint Conference on Artificial Intelligence, 2017).

On the generalized k-order additivity for absolutely monotone set functions

The concepts of the generalized Mobius transform and the generalized k-additivity for absolutely monotone and sign stable set functions are introduced and investigated. The evaluation formula for the discrete general Choquet-like integrals with respect to generalized k-order additive set functions is obtained.

Bi-directional dominance for measure modeled uncertainty

Our objective here is to provide a formulation for a dominance relationship between variables having uncertain values, represented by measures, which can be used for ordering these uncertain values. We first discuss the use of monotonic set functions, fuzzy measures, for the representation of uncertain information. We consider the issue of comparing and ordering variables whose values are uncertain and represented via a measure and we suggest the use of bi-directional dominance, which can be seen as a generalization of stochastic dominance that is used to order probability distributions. We note the establishment of a dominance relationship between uncertain values is often not possible. In order to circumvent this we introduce the idea of surrogates for bi-directional dominance, here we associate with each uncertain value, measure, a scalar value that provides an ordering between all uncertain values. These surrogates are defined so that they maintain any relationship between measures that can be established with the use of bi-directional dominance.

Optimal Approximation for Submodular and Supermodular Optimization with Bounded Curvature

We design new approximation algorithms for the problems of optimizing submodular and supermodular functions subject to a single matroid constraint. Specifically, we consider the case in which we wish to maximize a monotone increasing submodular function or minimize a monotone decreasing supermodular function with a bounded total curvature c. Intuitively, the parameter c represents how nonlinear a function f is: when c = 0, f is linear, while for c = 1, f may be an arbitrary monotone increasing submodular function. For the case of submodular maximization with total curvature c, we obtain a (1 − c/e)-approximation—the first improvement over the greedy algorithm of of Conforti and Cornuéjols from 1984, which holds for a cardinality constraint, as well as a recent analogous result for an arbitrary matroid constraint. Our approach is based on modifications of the continuous greedy algorithm and nonoblivious local search, and allows us to approximately maximize the sum of a nonnegative, monotone increasing submodular function and a (possibly negative) linear function. We show how to reduce both submodular maximization and supermodular minimization to this general problem when the objective function has bounded total curvature. We prove that the approximation results we obtain are the best possible in the value oracle model, even in the case of a cardinality constraint. We define an extension of the notion of curvature to general monotone set functions and show a (1 − c)-approximation for maximization and a 1/(1 − c)-approximation for minimization cases. Finally, we give two concrete applications of our results in the settings of maximum entropy sampling, and the column-subset selection problem.

A natural parametrization of multivariate distributions with limited memory

A unified formulation of the theory of d -variate wide-sense geometric ( G d W ) and Marshall–Olkin exponential ( MO d ) distributions is presented in which d -monotone set functions occupy a central role. A semi-analytical derivation of G d W and MO d distributions is deduced directly from the lack-of-memory property. In this context, the distributions are parametrized with d -monotone and d -log-monotone set functions arising from the univariate marginal distributions of minima and the d -decreasingness of the survival functions. In addition, a one-to-one correspondence is established between d -monotone (resp. d -log-monotone) set functions and d -variate (resp. d -variate min-infinitely divisible) Bernoulli distributions. The advantage of such a parametrization is that it makes the distributions highly tractable. As a showcase, we derive new results on the minimum stability and divisibility of the G d W family, and on the marginal equivalence in minima of G d W and distributions with geometric minima. Similarly, a surprisingly simple proof is given of the prominent result of Esary and Marshall (1974) on the marginal equivalence in minima of multivariate exponential distributions.

Integrals based on monotone set functions

An overview of various integrals is given which can be defined on arbitrary monotone set functions vanishing in the empty set (called here monotone measures). Our survey includes not only the Choquet integral (1954) [10], the Shilkret integral (1971) [66] and the Sugeno integral (1974) [71] and some of their properties, but also some more general and more recent concepts as universal integrals Klement et al. (2010) [27] and decomposition integrals Even (2014) [13], together with some of their properties, such as integral inequalities and convergence theorems.

A Unifying Hierarchy of Valuations with Complements and Substitutes

We introduce a new hierarchy over monotone set functions, that we refer to as MPH (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, MPH-m (where m is the total number of items) captures all monotone functions. The lowest level, MPH-1, captures all monotone submodular functions, and more generally, the class of functions known as XOS. Every monotone function that has a positive hypergraph representation of rank k (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in MPH-k. Every monotone function that has supermodular degree k (in the sense defined by Feige and Izsak [ITCS 2013]) is in MPH-(k+1). In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of MPH-k.One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the MPH hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of k+1 if all players hold valuation functions in MPH-k. The other is an upper bound of 2k on the price of anarchy of simultaneous first price auctions.

Exact bounds of the Möbius inverse of monotone set functions

We give the exact upper and lower bounds of the Möbius inverse of monotone and normalized set functions (a.k.a. normalized capacities) on a finite set of n elements. We find that the absolute value of the bounds tend to 4n/2πn/2 when n is large. We establish also the exact bounds of the interaction transform and Banzhaf interaction transform, as well as the exact bounds of the Möbius inverse for the subfamilies of k-additive normalized capacities and p-symmetric normalized capacities.

The Application of Conflict Measure to Estimating Incoherence of Analyst's Forecasts about the Cost of Shares of Russian Companies

Abstract This paper is devoted to modern approaches to the estimation of external conflict in the theory of evidence based on axioms. The conflict measure is defined on the set of beliefs obtained from several sources of information. It is shown that the conflict measure should be a monotone set function with respect to sets of beliefs. Some robust procedures for evaluation of conflict measure that are stable to small changes in evidences are introduced and discussed. The analysis of conflict among forecasts about the value of shares of Russian companies of investment banks is presented. In this analysis the conflict measure estimates inconsistency of recommendations of investment banks, while the Shapley values of this measure on the set of evidences characterize the contribution of each investment bank to the overall conflict. The relationship between conflict and precision of forecasts is also investigated.

Representing qualitative capacities as families of possibility measures

This paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite totally ordered scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multiple-criteria decision-making. We show that any capacity or fuzzy measure ranging on a qualitative scale can be viewed both as the lower bound of a set of possibility measures and the upper bound of a set of necessity measures (a situation somewhat similar to the one of quantitative capacities with respect to imprecise probability theory). We show that any capacity is characterized by a non-empty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and the number of them is characteristic of its complexity. An algorithm is provided to compute the minimal set of possibility measures dominating a given capacity. This algorithm relies on the representation of the capacity by means of its qualitative Möbius transform, and the use of selection functions of the corresponding focal sets. We provide the connection between Sugeno integrals and lower possibility measures. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular modal logics and their neighborhood semantics can be described in terms of qualitative Möbius transforms.

Stochastic dominance with respect to a capacity and risk measures

Abstract In our previous work, we have extended the classical notion of increasing convex stochastic dominance relation with respect to a probability to the more general case of a normalized monotone (but not necessarily additive) set function, also called a capacity. In the present paper, we pursue that work by studying the set of monetary risk measures (defined on the space of bounded real-valued measurable functions) satisfying the properties of comonotonic additivity and consistency with respect to the generalized stochastic dominance relation. Under suitable assumptions on the underlying capacity space, we characterize that class of risk measures in terms of Choquet integrals with respect to a distorted capacity whose distortion function is concave. Kusuoka-type characterizations are also established. A generalization to the case of a capacity of the Tail Value at Risk is provided as an example. It is also shown that some well-known results about Choquet integrals with respect to a distorted probability do not necessarily hold true in the more general case of a distorted capacity.

Stochastic orderings with respect to a capacity and an application to a financial optimization problem

Abstract By analogy with the classical case of a probability measure, we extend the notion of increasing convex (concave) stochastic dominance relation to the case of a normalized monotone (but not necessarily additive) set function also called a capacity. We give different characterizations of this relation establishing a link to the notions of distribution function and quantile function with respect to the given capacity. The Choquet integral is extensively used as a tool. In the second part of the paper, we give an application to a financial optimization problem whose constraints are expressed by means of the increasing convex stochastic dominance relation with respect to a capacity. The problem is solved by using, among other tools, a result established in our previous work, namely a new version of the classical upper (resp. lower) Hardy–Littlewood's inequality generalized to the case of a continuous from below concave (resp. convex) capacity. The value function of the optimization problem is interpreted in terms of risk measures (or premium principles).