Related papers: Algebraic Theory of Patterns as Generalized Symmet…
Symmetries are known to dictate important physical properties and can be used as a design principle in particular in wave physics, including wave structures and the resulting propagation dynamics. Local symmetries, in the sense of a…
The concept of sharpness has been successfully applied to traditional architectures like MLPs and CNNs to predict their generalization. For transformers, however, recent work reported weak correlation between flatness and generalization. We…
The author has recently introduced abstract algebraic frameworks of analogical proportions and similarity within the general setting of universal algebra. The purpose of this paper is to build a bridge from similarity to analogical…
In this paper we consider the problem of quantizing theories defined over configuration spaces described by non-commuting parameters. If one tries to do that by generalizing the path-integral formalism, the first problem one has to deal…
Group theory is extremely successful in characterizing the symmetries in quantum systems, which greatly simplifies and unifies our treatments of quantum systems. Here we introduce the concept of the symmetry for a quantum Boltzmann machine…
Recently, a geometrical characterization of vector spaces served to generalize them into a new class of algebras. Instead of the algebraic properties of the underlying fields, we generalized the recently discovered property of such spaces…
We construct a theory in which the gravitational interaction is described only by torsion, but that generalizes the Teleparallel Theory still keeping the invariance of local Lorentz transformations in one particular case. We show that our…
We consider coherent sublinear expectations on a measurable space, without assuming the existence of a dominating probability measure. By considering a decomposition of the space in terms of the supports of the measures representing our…
Stochastic models share many characteristics with generic parametric models. In some ways they can be regarded as a special case. But for stochastic models there is a notion of weak distribution or generalised random variable, and the same…
Any representation of data involves arbitrary investigator choices. Because those choices are external to the data-generating process, each choice leads to an exact symmetry, corresponding to the group of transformations that takes one…
We introduce semiframes (an algebraic structure) and investigate their duality with semitopologies (a topological one). Both semitopologies and semiframes are relatively recent developments, arising from a novel application of topological…
We study the approximation properties of convolutional architectures applied to time series modelling, which can be formulated mathematically as a functional approximation problem. In the recurrent setting, recent results reveal an…
Discovering pragmatic and efficient approaches to construct $\varepsilon$-approximations of quantum operators such as real (imaginary) time-evolution propagators in terms of the basic quantum operations (gates) is challenging. Prior…
Many problems in machine learning involve calculating correspondences between sets of objects, such as point clouds or images. Discrete optimal transport provides a natural and successful approach to such tasks whenever the two sets of…
We prove that each semialgebraic subset of $\R^n$ of positive codimension can be locally approximated of any order by means of an algebraic set of the same dimension. As a consequence of previous results, algebraic approximation preserving…
The recollement approach to the representation theory of sequences of algebras is extended to pass basis information directly through the globalisation functor. The method is hence adapted to treat sequences that are not necessarily towers…
We examine generalized global symmetries as a kind of compactly supported cohomology, and so are led to revisit questions about the locality of quantum field theory, following Segal. Physics naturally suggests a generalization of…
We discuss the homological algebra of representation theory of finite dimensional algebras and finite groups. We present various methods for the construction and the study of equivalences of derived categories: local group theory, geometry…
We give characterizations of unital uniform topological algebras and saturated locally multiplicatively convex algebras by means of multiplicative linear functionals. Some automatic continuity theorems in advertibly complete uniform…
Profinite semigroups are a generalization of finite semigroups that come about naturally when one is interested in considering free structures with respect to classes of finite semigroups. They also appear naturally through dualization of…