Related papers: Near approximations via general ordered topologica…
The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets. Systems of equations encoded by complex trees tip-to-tip equivalence relations are used to obtain one-parameter families of connected…
Generalization is at the core of machine learning models. However, the definition of generalization is not entirely clear. We employ set theory to introduce the concepts of algorithms, hypotheses, and dataset generalization. We analyze the…
Complex prediction models such as deep learning are the output from fitting machine learning, neural networks, or AI models to a set of training data. These are now standard tools in science. A key challenge with the current generation of…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. Starting from ZFC, the exposition in this first part includes relation and order theory as well as a construction of…
The universal approximation theorem asserts that a single hidden layer neural network approximates continuous functions with any desired precision on compact sets. As an existential result, the universal approximation theorem supports the…
We introduce and study a generalized concept of boundedness of a subset of a normed vector space with respect to a cone, which is defined as lower boundedness of the images of the underlying set through all the positive functionals of the…
Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration…
This article delves into the study of the theory of regularized learning in Banach spaces for linear-functional data. It encompasses discussions on representer theorems, pseudo-approximation theorems, and convergence theorems. Regularized…
Two aspects of neural networks that have been extensively studied in the recent literature are their function approximation properties and their training by gradient descent methods. The approximation problem seeks accurate approximations…
Graph-based frames have been introduced as a logical framework which internalizes an inherent boundary to knowability. They also support the interpretation of lattice-based (modal) logics as hyper-constructive logics of evidential…
Computational visual aesthetics has recently become an active research area. Existing state-of-art methods formulate this as a binary classification task where a given image is predicted to be beautiful or not. In many applications such as…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
Modularity is a very widely used measure of the level of clustering or community structure in networks. Here we consider a recent generalisation of the definition of modularity to temporal graphs, whose edge-sets change over discrete…
The basics of set theory are usually copied, directly or indirectly, by computer scientists from introductions to mathematical texts. Often mathematicians are content with special cases when the general case is of no mathematical interest.…
Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and…
One of the basic principles of Approximation Theory is that the quality of approximations increase with the smoothness of the function to be approximated. Functions that are smooth in certain subdomains will have good approximations in…
An ordered $r$-matching is an $r$-uniform hypergraph matching equipped with an ordering on its vertices. These objects can be viewed as natural generalisations of $r$-dimensional orders. The theory of ordered 2-matchings is well-developed…
The relationship according to which one physical theory encompasses the domain of empirical validity of another is widely known as "reduction." Here it is argued that one popular methodology for showing that one theory reduces to another,…
Several real-world and abstract structures and systems are characterized by marked hierarchy to the point of being expressed as trees. Because the study of these entities often involves sampling (or discovering) the tree nodes in a specific…
Reachable set computation is an important tool for analyzing control systems. Simulating a control system can show general trends, but a formal tool like reachability analysis can provide guarantees of correctness. Reachability analysis for…