Related papers: Continuous selections of multivalued mappings
Multimodal classification research has been gaining popularity in many domains that collect more data from multiple sources including satellite imagery, biometrics, and medicine. However, the lack of consistent terminology and architectural…
This habilitation thesis summarizes the research that I have carried out from 2005 to 2019. It is organized in four chapters. The first three deal with random planar maps. Chapter 1 is about their metric properties: from a general…
This survey focuses on the most important aspects of the mathematical theory of population genetic models of selection and migration between discrete niches. Such models are most appropriate if the dispersal distance is short compared to…
As suggested by the title, this paper is a survey of recent results and questions on the collection of computably enumerable sets under inclusion. This is not a broad survey but one focused on the author's and a few others' current…
In 20th century mathematics, the field of topology, which concerns the properties of geometric objects under continuous transformation, has proved surprisingly useful in application to the study of discrete mathematics, such as…
Topological Data Analysis (TDA) is a recent and growing branch of statistics devoted to the study of the shape of the data. In this work we investigate the predictive power of TDA in the context of supervised learning. Since topological…
In work started in [17] and continued in this paper our objective is to study selectors of multivalued functions which have interesting dynamical properties, such as possessing absolutely continuous invariant measures. We specify the graph…
Deep learning is usually described as an experiment-driven field under continuous criticizes of lacking theoretical foundations. This problem has been partially fixed by a large volume of literature which has so far not been well organized.…
This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or…
This article presents a survey of some recent results in the theory of spatial graphs. In particular, we highlight results related to intrinsic knotting and linking and results about symmetries of spatial graphs. In both cases we consider…
This is an overview of the area of Stochastic Portfolio Theory, and can be seen as an updated and extended version of the survey paper by Fernholz and Karatzas (Handbook of Numerical Analysis Vol.15:89-167, 2009).
Clustering is a fundamental machine learning task which has been widely studied in the literature. Classic clustering methods follow the assumption that data are represented as features in a vectorized form through various representation…
We develop Conley's theory for multivalued maps on finite topological spaces. More precisely, for discrete-time dynamical systems generated by the iteration of a multivalued map which satisfies appropriate regularity conditions, we…
We prove that Michael's paraconvex-valued selection theorem for paracompact spaces remains true for C'(E)-valued mappings defined on collectionwise normal spaces. Some possible generalisations are also given.
In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators of persistent homology. This estimation…
The main purpose of this paper is to extend some fixed point results for single valued $b$-enriched nonexpansive mappings to the case of multivalued mappings. To this end, we introduce *-$b$-enriched nonexpansive mappings, as a…
This survey contains the main results in rational homotopy, from the beginning to the most recent ones. It makes the status of the art, gives a short presentation of some areas where rational homotopy has been used, and contains a lot of…
An overview of some of the recent developments in the theory of valuations on convex sets and its generalizations to manifolds is given. The exposition is focused towards applications to integral geometry; several of such applications are…
Deep Learning has shown great success in reshaping medical imaging, yet it faces numerous challenges hindering widespread application. Issues like catastrophic forgetting and distribution shifts in the continuously evolving data stream…
Grouping structures arise naturally in many statistical modeling problems. Several methods have been proposed for variable selection that respect grouping structure in variables. Examples include the group LASSO and several concave group…