The transport problem for non-additive measures

Non-additive measures, also known as fuzzy measures, capacities, and monotonic games, are increasingly used in different fields. Applications have been built within computer science and artificial intelligence related to e.g. decision making, image processing, machine learning for both classification, and regression. Tools for measure identification have been built. In short, as non-additive measures are more general than additive ones (i.e., than probabilities), they have better modeling capabilities allowing to model situations and problems that cannot be modeled by the latter. See e.g. the application of non-additive measures and the Choquet integral to model both Ellsberg paradox and Allais paradox. Because of that, there is an increasing need to analyze non-additive measures. The need for distances and similarities to compare them is no exception. Some work has been done for defining $f$-divergence for them. In this work we tackle the problem of defining the optimal transport problem for non-additive measures. Distances for pairs of probability distributions based on the optimal transport are extremely used in practical applications, and they are being studied extensively for their mathematical properties. We consider that it is necessary to provide appropriate definitions with a similar flavour, and that generalize the standard ones, for non-additive measures. We provide definitions based on the M\"obius transform, but also based on the $(\max, +)$-transform that we consider that has some advantages. We will discuss in this paper the problems that arise to define the transport problem for non-additive measures, and discuss ways to solve them. In this paper we provide the definitions of the optimal transport problem, and prove some properties.

Literature Review of various Fuzzy Rule based Systems

Fuzzy rule based systems (FRBSs) is a rule-based system which uses linguistic fuzzy variables as antecedents and consequent to represent the human understandable knowledge. They have been applied to various applications and areas throughout the literature. However, FRBSs suffers from many drawbacks such as uncertainty representation, high number of rules, interpretability loss, high computational time for learning etc. To overcome these issues with FRBSs, there exists many extentions of FRBSs. In this paper, we present an overview and literature review for various types and prominent areas of fuzzy systems (FRBSs) namely genetic fuzzy system (GFS), Hierarchical fuzzy system (HFS), neuro fuzzy system (NFS), evolving fuzzy system (eFS), FRBSs for big data, FRBSs for imbalanced data, interpretability in FRBSs and FRBSs which uses cluster centroids as fuzzy rule, during the years 2010-2021. GFS uses genetic/evolutionary approaches to improve the learning ability of FRBSs, HFS solve the curse of dimensionality for FRBSs, NFS improves approximation ability of FRBSs using neural networks and dynamic systems for streaming data is considered in eFS. FRBSs are seen as good solutions for big data and imbalanced data, in the recent years the interpretability in FRBSs has gained popularity due to high dimensional and big data and rules are initialized with cluster centroids to limit the number of rules in FRBSs. This paper also highlights important contributions, publication statistics and current trends in the field. The paper also addresses several open research areas which need further attention from the FRBSs research community.

A formalization of re-identification in terms of compatible probabilities

Re-identification algorithms are used in data privacy to measure disclosure risk. They model the situation in which an adversary attacks a published database by means of linking the information of this adversary with the database. In this paper we formalize this type of algorithm in terms of true probabilities and compatible belief functions. The purpose of this work is to leave aside as re-identification algorithms those algorithms that do not satisfy a minimum requirement.