Related papers: Any-Dimensional Invariant Universality

Group symmetry is inherent in a wide variety of data distributions. Data processing that preserves symmetry is described as an equivariant map and often effective in achieving high performance. Convolutional neural networks (CNNs) have been…

Considering smooth mappings from input vectors to continuous targets, our goal is to characterise subspaces of the input domain, which are invariant under such mappings. Thus, we want to characterise manifolds implicitly defined by level…

An oblivious mobile robot is a stateless computational entity located in a spatial universe, capable of moving in that universe. When activated, the robot observes the universe and the location of the other robots, chooses a destination,…

A practical limitation of deep neural networks is their high degree of specialization to a single task and visual domain. Recently, inspired by the successes of transfer learning, several authors have proposed to learn instead universal,…

The problem of uniqueness of universal formulae for (quantum) dimensions of simple Lie algebras is investigated. We present generic functions, which multiplied by a universal (quantum) dimension formula, preserve both its structure and its…

Correctly capturing the symmetry transformations of data can lead to efficient models with strong generalization capabilities, though methods incorporating symmetries often require prior knowledge. While recent advancements have been made…

We introduce the notion of universal memcomputing machines (UMMs): a class of brain-inspired general-purpose computing machines based on systems with memory, whereby processing and storing of information occur on the same physical location.…

Nonlinear models and optimization methods have successfully tackled a rapidly growing set of problems in recent years. Indeed, a relatively small toolbox of such models and methods can provide sufficient performance across a large landscape…

This work suggests using sampling theory to analyze the function space represented by neural networks. First, it shows, under the assumption of a finite input domain, which is the common case in training neural networks, that the function…

We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to…

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…

In this paper we try to organize machine teaching as a coherent set of ideas. Each idea is presented as varying along a dimension. The collection of dimensions then form the problem space of machine teaching, such that existing teaching…

We establish universal approximation theorems for infinite-dimensional geometric rough paths, i.e., we show that continuous functions on the space of infinite-dimensional weakly geometric H\"older continuous rough paths can be approximated…

This article presents a theoretical investigation of generalized encoded forms of networks in a uniform multidimensional space. First, we study encoded networks with (finite) arbitrary node dimensions (or aspects), such as time instants or…

Large language models are capable of in-context learning, the ability to perform new tasks at test time using a handful of input-output examples, without parameter updates. We develop a universal approximation theory to elucidate how…

Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to…

We construct a universal space for the class of proper metric spaces of bounded geometry and of given asymptotic dimension. As a consequence of this result, we establish coincidence of the asymptotic dimension with the asymptotic inductive…

The concept of data depth leads to a center-outward ordering of multivariate data, and it has been effectively used for developing various data analytic tools. While different notions of depth were originally developed for finite…

Machine Learning is a diverse field applied across various domains such as computer science, social sciences, medicine, chemistry, and finance. This diversity results in varied evaluation approaches, making it difficult to compare models…

This paper investigates spaces equipped with a family of metric-like functions satisfying certain axioms. These functions provide a unified framework for defining topology, uniformity, and diffeology. The framework is based on a family of…