Related papers: CantorNet: A Sandbox for Testing Geometrical and T…

We study the computational problem of rigorously describing the asymptotic behaviour of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit,…

A 1910 theorem of Brouwer characterizes the Cantor set as the unique totally disconnected, compact metric space without isolated points. A 1920 theorem of Sierpinski characterizes the rationals as the unique countable metric space without…

In this paper, we study the Hausdorff dimension of self-similar measures and sets on the real line, where the generating iterated function system consists of some maps that share the same fixed point. In particular, we will show that out of…

We study some variants of the Erd\H{o}s similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset…

A methodology based upon recurrence quantification analysis is proposed for the study of orthographic structure of written texts. Five different orthographic data sets (20th century Italian poems, 20th century American poems, contemporary…

In this paper we obtain the rates of convergence of the algorithms given in [13] and [14] for an automatic computation of the centered Hausdorff and packing measures of a totally disconnected self-similar set. We evaluate these rates…

The objective of this work is set-based verification, e.g. to decide if two sets of images of a face are of the same person or not. The traditional approach to this problem is to learn to generate a feature vector per image, aggregate them…

A topological neural network (TNN), which takes data from a Tychonoff topological space instead of the usual finite dimensional space, is introduced. As a consequence, a distributional neural network (DNN) that takes Borel measures as data…

We construct a geometrically self-similar Cantor set $X$ of genus $2$ in $\mathbb{R}^3$. This construction is the first for which the local genus is shown to be $2$ at every point of $X$. As an application, we construct, also for the first…

Quantification of symmetries in complex networks is typically done globally in terms of automorphisms. Extending previous methods to locally assess the symmetry of nodes is not straightforward. Here we present a new framework to quantify…

We report an approach to obtaining complex networks with diverse topology, here called syntonets, taking into account the consonances and dissonances between notes as defined by scale temperaments. Though the fundamental frequency is…

This paper provides a new similarity detection algorithm. Given an input set of multi-dimensional data points, where each data point is assumed to be multi-dimensional, and an additional reference data point for similarity finding, the…

In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show…

Unsupervised point cloud shape correspondence aims to obtain dense point-to-point correspondences between point clouds without manually annotated pairs. However, humans and some animals have bilateral symmetry and various orientations,…

Despite significant progress of deep learning in recent years, state-of-the-art semantic matching methods still rely on legacy features such as SIFT or HoG. We argue that the strong invariance properties that are key to the success of…

Deep learning models are often considered black boxes due to their complex hierarchical transformations. Identifying suitable architectures is crucial for maximizing predictive performance with limited data. Understanding the geometric…

This paper introduces a new similarity measure based on edge counting in a taxonomy like WorldNet or Ontology. Measurement of similarity between text segments or concepts is very useful for many applications like information retrieval,…

Complex networks are a powerful modeling tool, allowing the study of countless real-world systems. They have been used in very different domains such as computer science, biology, sociology, management, etc. Authors have been trying to…

Point cloud processing poses two fundamental challenges: establishing consistent point ordering and effectively learning fine-grained geometric features. Current architectures rely on complex operations that limit expressivity while…

Advances in experimental neuroscience have transformed our ability to explore the structure and function of neural circuits. At the same time, advances in machine learning have unleashed the remarkable computational power of artificial…