Related papers: The geometry of learning
We introduce two novel families of geometric functionals-basic contents and support contents-for investigating the fractal properties of compact subsets in Euclidean space. These functionals are derived from the support measures arising in…
This article discusses the notion of convergence of sequences of iterated function systems. The technique of iterated function systems is one of the several methods to construct objects with fractal nature, and the fractals obtained with…
When plane waves diffract through fractal-patterned apertures, the resulting far-field profiles or diffractals also exhibit iterated, self-similar features. Here we show that this specific architecture enables robust signal processing and…
The fractal or Hausdorff dimension is a measure of roughness (or smoothness) for time series and spatial data. The graph of a smooth, differentiable surface indexed in $\mathbb{R}^d$ has topological and fractal dimension $d$. If the surface…
We define a class of random measures, spatially independent martingales, which we view as a natural generalisation of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian…
Fundamental limits to predictability are central to our understanding of many physical and computational systems. Here we show that, despite its remarkable capabilities, deep learning exhibits such fundamental limits rooted in the fractal,…
We propose a novel combinatorial algorithm for efficient generation of Hamiltonian walks and cycles on a cubic lattice, modeling the conformations of lattice toy proteins. Through extensive tests on small lattices (allowing complete…
We discuss a model of protein conformations where the conformations are combinations of short fragments from some small set. For these fragments we consider a distribution of frequencies of occurrence of pairs (sequence of amino acids,…
The foundation of the theory presented here has already been proved to be effective for the case of curves belonging to the Koch family. The present paper extends the investigation to more complex curves, namely randomly generated curves…
This is a survey paper about the fractal percolation process, also known as Mandelbrot percolation. It is intended to give a general breadth overview of more recent research in the topic, but also includes some of the more classical…
Developing a robust generalization measure for the performance of machine learning models is an important and challenging task. A lot of recent research in the area focuses on the model decision boundary when predicting generalization. In…
Disobeying the classical wisdom of statistical learning theory, modern deep neural networks generalize well even though they typically contain millions of parameters. Recently, it has been shown that the trajectories of iterative…
The two-fractal overlap model of earthquake shows that the contact area distribution of two fractal surfaces follows power law decay in many cases and this agrees with the Guttenberg-Richter power law. Here, we attempt to predict the large…
Fractal structure of the six-vertex model is introduced with the use of the IFS (Iterated Function Systems). The fractal dimension satisfies an equation written by the free energy of the six-vertex model. It is pointed out that the transfer…
Fractal geometries, characterized by self-similar patterns and non-integer dimensions, provide an intriguing platform for exploring topological phases of matter. In this work, we introduce a theoretical framework that leverages isospectral…
The improved city clustering algorithm can be used to identify urban boundaries on a digital map, and the results are a set of isolines. The relationships between the urban measurements within the variable boundaries follow allometric…
This paper surveys work on the relation between fractal dimensions and algorithmic information theory over the past thirty years. It covers the basic development of prefix-free Kolmogorov complexity from an information theoretic point of…
Urban form has been empirically demonstrated to be of scaling invariance and can be described with fractal geometry. However, the rational range of fractal dimension value and the relationships between various fractal indicators of cities…
In this paper, we introduce the concept of the $\alpha$-fractal function and fractal approximation for a set-valued continuous map defined on a closed and bounded interval of real numbers. Also, we study some properties of such fractal…
Although geographic features, such as mountains and coastlines, are fractal, some studies have claimed that the fractal property is not universal. This claim, which is false, is mainly attributed to the strict definition of fractal…