Complementary Lipschitz continuity results for the distribution of intersections or unions of independent random sets in finite discrete spaces

We prove that intersections and unions of independent random sets in finite spaces achieve a form of Lipschitz continuity. More precisely, given the distribution of a random set $\Xi$, the function mapping any random set distribution to the distribution of its intersection (under independence assumption) with $\Xi$ is Lipschitz continuous with unit Lipschitz constant if the space of random set distributions is endowed with a metric defined as the $L_k$ norm distance between inclusion functionals also known as commonalities. Moreover, the function mapping any random set distribution to the distribution of its union (under independence assumption) with $\Xi$ is Lipschitz continuous with unit Lipschitz constant if the space of random set distributions is endowed with a metric defined as the $L_k$ norm distance between hitting functionals also known as plausibilities. Using the epistemic random set interpretation of belief functions, we also discuss the ability of these distances to yield conflict measures. All the proofs in this paper are derived in the framework of Dempster-Shafer belief functions. Let alone the discussion on conflict measures, it is straightforward to transcribe the proofs into the general (non necessarily epistemic) random set terminology

Knowability as continuity: a topological account of informational dependence

We study knowable informational dependence between empirical questions, modeled as continuous functional dependence between variables in a topological setting. We also investigate epistemic independence in topological terms and show that it is compatible with functional (but non-continuous) dependence. We then proceed to study a stronger notion of knowability based on uniformly continuous dependence. On the technical logical side, we determine the complete logics of languages that combine general functional dependence, continuous dependence, and uniformly continuous dependence.Comment: 65 page

Topological and geometric inference of data

The overarching problem under consideration is to determine the structure of the subspace on which a distribution is supported, given only a finite noisy sample thereof. The special case in which the subspace is an embedded manifold is given particular attention owing to its conceptual elegance, and asymptotic bounds are obtained on the admissible level of noise such that the manifold can be recovered up to homotopy equivalence. Attention is turned on how to accomplish this in practice. Following ideas from topological data analysis, simplicial complexes are used as discrete analogues of spaces suitable for computation. By utilising the prior assumption that the data lie on a manifold, topologically inspired techniques are proposed for refining the simplicial complex to better approximate this manifold. This is applied to the problem of nonlinear dimensionality reduction and found to improve accuracy of reconstructing several synthetic and real-world datasets. The second chapter focuses on extending this work to the case where the ambient space is non-Euclidean. The interfaces between topological data analysis, functional data analysis, and shape analysis are thoroughly explored. Lipschitz bounds are proved which relate several metrics on the space of positive semidefinite matrices; they are then interpreted in the context of topological data analysis. This is applied to diffusion tensor imaging and phonology. The final chapter explores the case where the points are non-uniformly distributed over the embedded subspace. In particular, a method is proposed to overcome the shortcomings of witness complex construction when there are large deviations in the density. The theory of multidimensional persistence is leveraged to provide a succinct setting in which the structure of the data can be interpreted as a generalised stratified space.EPSR

New Directions in Geometric and Applied Knot Theory

The aim of this book is to present recent results in both theoretical and applied knot theory—which are at the same time stimulating for leading researchers in the field as well as accessible to non-experts. The book comprises recent research results while covering a wide range of different sub-disciplines, such as the young field of geometric knot theory, combinatorial knot theory, as well as applications in microbiology and theoretical physics

STUDIES ON STOCHASTIC ALGORITHMS INFORMATION CONTENT, PARALLELISATION AND DIFFUSION INDUCED STOCHASTIC ALGORITHMS FOR GLOBAL OPTIMISATION

This thesis presents the main results of two articles published by the authors in the field of stochastic optimization. We dedicated the chapter 1 to the article introduction to the information content of some stochastic algorithms written by Esquível, Machado, Krasii, and Mota, 2021. In this chapter, we formulate an optimization stochastic algorithm convergence theorem, of Solis and Wets type, and we show several instances of its application to concrete algorithms. In this convergence theorem the algorithm is a sequence of random variables and, in order to describe the increasing flow of information associated to this sequence we define a filtration – or flow of σ-algebras – on the probability space, depending on the sequence of random variables and on the function being optimized. We compare the flow of information of two convergent algorithms by comparing the associated filtrations by means of the Cotter distance of σ-algebras. The main result is that two convergent optimization algorithms have the same information content if both their limit minimization functions generate the full σ-algebra of the probability space. The article On a Parallelised Diffusion Induced Stochastic Algorithm with Pure Random Search Steps for Global Optimisation written by Esquível, Krasii, Mota, and Machado, 2021 was broken down into 2 chapters: the chapter 2 is related to parallelisation and the chapter 3 is related to Diffusion Induced Stochastic Algorithms. In the chapter 2 we show that an adequate procedure of parallelisation of the algorithm can increase the rate of convergence, thus superseding the main drawback of the addition of the pure random search step. Finally, in the chapter 3 we propose a stochastic algorithm for global optimisation of a regular function, possibly unbounded, defined on a bounded set with regular boundary; a function that attains its extremum in the boundary of its domain of definition. The algorithm is determined by a diffusion process that is associated with the function by means of a strictly elliptic operator that ensures an adequate maximum principle. In order to preclude the algorithm to be trapped in a local extremum, we add a pure random search step to the algorithm. As the two articles have their own introductions, we decided to create a glossary that together with the annexes and appendices, include the concepts, definitions and theorems that are relevant to the understanding of the thesis

Basic Probability Theory

Long title: Basic Probability Theory: Independent Random Variables and Sample Spaces. Chapters: Elementary Probability - Basic Probability - Canonical Sample Spaces - Working on Probability Spaces - A Solutions to Exercises