Related papers: CantorNet: A Sandbox for Testing Geometrical and T…
Given $\lambda\in(0, 1)$, let $E_\lambda$ be the self-similar set generated by the iterated function system $\{x/3,(x+\lambda)/3,(x+2)/3\}$. Then $E_\lambda$ is a self-similar set with overlaps. We obtain the necessary and sufficient…
Mathematical formulas serve as the means of communication between humans and nature, encapsulating the operational laws governing natural phenomena. The concise formulation of these laws is a crucial objective in scientific research and an…
The main aim of the paper is to introduce a new class of (semigroup-valued) measures that are ultrahomogeneous on the Boolean algebra of all clopen subsets of the Cantor space and to study their automorphism groups. A characterisation, in…
Recurrence networks are a powerful nonlinear tool for time series analysis of complex dynamical systems. {While there are already many successful applications ranging from medicine to paleoclimatology, a solid theoretical foundation of the…
This paper proposes a novel paradigm for machine learning that moves beyond traditional parameter optimization. Unlike conventional approaches that search for optimal parameters within a fixed geometric space, our core idea is to treat the…
We consider fixed-point equations for probability measures charging measured compact metric spaces that naturally yield continuum random trees. On the one hand, we study the existence/uniqueness of the fixed-points and the convergence of…
Considerable efforts in modern statistical physics is devoted to the study of networked systems. One of the most important example of them is the brain, which creates and continuously develops complex networks of correlated dynamics. An…
We introduce a topological object, called hairy Cantor set, which in many ways enjoys the universal features of objects like Jordan curve, Cantor set, Cantor bouquet, hairy Jordan curve, etc. We give an axiomatic characterisation of hairy…
The goal of this paper is to analyze the geometric properties of deep neural network classifiers in the input space. We specifically study the topology of classification regions created by deep networks, as well as their associated decision…
Can music be represented as a meaningful geometric and topological object? In this paper, we propose a strategy to describe some music features as a polyhedral surface obtained by a simplicial interpretation of the \textit{Tonnetz}. The…
Studies show that neural networks, not unlike traditional programs, are subject to bugs, e.g., adversarial samples that cause classification errors and discriminatory instances that demonstrate the lack of fairness. Given that neural…
The study of neural computation aims to understand the function of a neural system as an information processing machine. Neural systems are undoubtedly complex, necessitating principled and automated tools to abstract away details to…
Despite their success, modern convolutional neural networks (CNNs) exhibit fundamental limitations, including data inefficiency, poor out-of-distribution generalization, and vulnerability to adversarial perturbations. These shortcomings can…
This paper presents MONET -- an end-to-end semi-supervised learning framework for a keypoint detector using multiview image streams. In particular, we consider general subjects such as non-human species where attaining a large scale…
We define the concept of self-similarity of an object by considering endomorphisms of the object as `similarity' maps. A variety of interesting examples of self-similar objects in geometry, algebra and arithmetic are introduced.…
We describe new approaches for distances between pairs of 2-dimensional surfaces (embedded in 3-dimensional space) that use local structures and global information contained in inter-structure geometric relationships. We present algorithms…
Machine learning applications cover a wide range of predictive tasks in which tabular datasets play a significant role. However, although they often address similar problems, tabular datasets are typically treated as standalone tasks. The…
In order to design haptic icons or build a haptic vocabulary, we require a set of easily distinguishable haptic signals to avoid perceptual ambiguity, which in turn requires a way to accurately estimate the perceptual (dis)similarity of…
Topological materials exhibit unique electronic structures that underpin both fundamental quantum phenomena and next-generation technologies, yet their discovery remains constrained by the high computational cost of first-principles…
Finding self-similarity is a key step for understanding the governing law behind complex physical phenomena. Traditional methods for identifying self-similarity often rely on specific models, which can introduce significant bias. In this…