Related papers: Machine learning and the Continuum Hypothesis

In this paper, we introduce and develop the concept of conditional quantization for Borel probability measures on $\mathbb{R}^k,$ considering both constrained and unconstrained frameworks. For each setting, we define the associated…

Approximation theory has long been concerned with the development of positive linear operators that effectively approximate classes of functions. Among the most well-known results in this area are Korovkin-type approximation theorems, which…

It has been demonstrated earlier that universal computation is 'almost surely' chaotic. Machine learning is a form of computational fixed point iteration, iterating over the computable function space. We showcase some properties of this…

Learning from non-stationary data streams and overcoming catastrophic forgetting still poses a serious challenge for machine learning research. Rather than aiming to improve state-of-the-art, in this work we provide insight into the limits…

The aim of this paper is to prove ergodic decomposition theorems for probability measures quasi-invariant under Borel actions of inductively compact groups (Theorem 1) as well as for sigma-finite invariant measures (Corollary 1). For…

We develop Morse-Bott theory on posets, generalizing both discrete Morse-Bott theory for regular complexes and Morse theory on posets. Moreover, we prove a Lusternik-Schnirelmann theorem for general matchings on posets, in particular, for…

We present a learning theory for the training of a linear system operator having an input compositional variable and propose a Bayesian inversion method for inferring the unknown variable from an output of a noisy linear system. We assume…

Kuratowski's 14-set theorem says that in a topological space, 14 is the maximum possible number of distinct sets which can be generated from a fixed set by taking closures and complements. In this article we consider the analogous questions…

We provide a pointwise bipolar theorem for liminf-closed convex sets of positive Borel measurable functions on a sigma-compact metric space without the assumption that the polar is a tight set of measures. As applications we derive a…

Set prediction is about learning to predict a collection of unordered variables with unknown interrelations. Training such models with set losses imposes the structure of a metric space over sets. We focus on stochastic and underdefined…

The traditional approach of statistical physics to supervised learning routinely assumes unrealistic generative models for the data: usually inputs are independent random variables, uncorrelated with their labels. Only recently, statistical…

We present some results and conjectures on a generalization to the noncommutative setup of the Brouwer fixed-point theorem from the Borsuk-Ulam theorem perspective.

Set function learning has emerged as a crucial area in machine learning, addressing the challenge of modeling functions that take sets as inputs. Unlike traditional machine learning that involves fixed-size input vectors where the order of…

The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In…

In semantics and in programming practice, algebraic concepts such as monads or, essentially equivalently, (large) Lawvere theories are a well-established tool for modelling generic side-effects. An important issue in this context are…

Consider the lattice of bounded linear operators on the space of Borel measures on a Polish space. We prove that the operators which are continuous with respect to the weak topology induced by the bounded measurable functions form a…

While automata theory often concerns itself with regular predicates, relations corresponding to acceptance by a finite state automaton, in this article we study the regular functions, such relations which are also functions in the…

There are certain countably generated sigma-algebras of sets in the real line which do not admit any non-zero, sigma-finite, diffused (or, continuous) measure. Such countably generated sigma-algebras can be obtained by the use of some…

Here it is shown that standard set theory can be interpreted in a theory about order. The ordering here is about non-extensional flat classes, i.e. classes that are not elements of classes. So, stipulating a nearly well order over all those…

This paper studies recurrence phenomena in iterative holomorphic dynamics of certain multi-valued maps. In particular, we prove an analogue of the Poincar\'e recurrence theorem for meromorphic correspondences with respect to certain…