Related papers: Interpolating between dimensions
\emph{Fractal percolation} or \emph{Mandelbrot percolation} is one of the most well studied families of random fractals. In this paper we study some of the geometric measure theoretical properties (dimension of projections and structure of…
We relate various concepts of fractal dimension of the limiting set C in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in C (the "dust"). In two dimensions, we…
We estimate a Box-counting dimension of fractal surfaces which are generated by iterated function systems with a vertical contraction factor function on an arbitrary data set over rectangular grids and can express well a lot of natural…
We introduce a fractal dimension for a metric space defined in terms of the persistent homology of extremal subsets of that space. We exhibit hypotheses under which this dimension is comparable to the upper box dimension; in particular, the…
A prescription is presented for the interpolation between multi-dimensional distribution templates based on one or multiple model parameters. The technique uses a linear combination of templates, each created using fixed values of the…
The area-perimeter scaling can be employed to evaluate the fractal dimension of urban boundaries. However, the formula in common use seems to be not correct. By means of mathematical method, a new formula of calculating the boundary…
We propose that the recently defined persistent homology dimensions are a practical tool for fractal dimension estimation of point samples. We implement an algorithm to estimate the persistent homology dimension, and compare its performance…
The notions of distance and similarity play a key role in many machine learning approaches, and artificial intelligence (AI) in general, since they can serve as an organizing principle by which individuals classify objects, form concepts…
We introduce a new family of fractal dimensions by restricting the set of diameters in the coverings in the usual definition of the Hausdorff dimension. Among others, we prove that this family contains continuum many distinct dimensions,…
Fractal dimension is defined on the base of entropy, including macro state entropy and information entropy. The generalized correlation dimension of multifractals is based on Renyi entropy. However, the mathematical transform from entropy…
Bounding and predicting the generalization gap of overparameterized neural networks remains a central open problem in theoretical machine learning. There is a recent and growing body of literature that proposes the framework of fractals to…
The theory of thick distributions (both in dimension 1 and in higher dimensions) was constructed in recent years [7, 27]. However this theory of distributions with one thick point in dimension one is very different from that in higher…
I present a framework based on the concepts of diversity and coherence for the analysis of knowledge integration and diffusion. Visualisations that help understand insights gained are also introduced. The key novelty offered by this…
The Interstellar Medium seems to have an underlying fractal structure, which can be characterized through its fractal dimension (Df). However, several factors may affect the determination of Df, such as distortions due to projection, low…
The structure of the large scale distribution of the galaxies have been widely studied since the publication of the first catalogs. Since large redshift samples are available, their analyses seem to show fractal correlations up to the…
In this short note, we merge the areas of hypercomplex algebras with that of fractal interpolation and approximation. The outcome is a new holistic methodology that allows the modelling of phenomena exhibiting a complex self-referential…
A rigorous mathematical theory of dimensional analysis, systematically accounting for the use of physical quantities in science and engineering, perhaps surprisingly, was not developed until relatively recently. We claim that this has…
Non-autonomous iterated function systems are a generalization of iterated function systems. If the contractions in the system are conformal mappings, it is called a non-autonomous conformal iterated function system, and its attractor is…
We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension…
Our goal in this paper is to develop an effective estimator of fractal dimension. We survey existing ideas in dimension estimation, with a focus on the currently popular method of Grassberger and Procaccia for the estimation of correlation…