Related papers: Fractal dimension, approximation and data sets
Principal component analysis (PCA) has well-documented merits for data extraction and dimensionality reduction. PCA deals with a single dataset at a time, and it is challenged when it comes to analyzing multiple datasets. Yet in certain…
Fractal dimensions have been used as a quantitative measure for structure of eigenstates of quantum many-body systems, useful for comparison to random matrix theory predictions or to distinguish many-body localized systems from chaotic…
Improvements in computational and experimental capabilities are rapidly increasing the amount of scientific data that is routinely generated. In applications that are constrained by memory and computational intensity, excessively large…
We focus on mesoscopic dislocation patterning via a continuum dislocation dynamics theory (CDD) in three dimensions (3D). We study three distinct physically motivated dynamics which consistently lead to fractal formation in 3D with rather…
In the interstellar medium, as well as in the Universe, large density fluctuations are observed, that obey power-law density distributions and correlation functions. These structures are hierarchical, chaotic, turbulent, but are also…
We study a variant of the Falconer distance problem for dot products. In particular, for fractal subsets $A\subset \mathbb{R}^n$ and $a,x\in \mathbb{R}^n$, we study sets of the form \[ \Pi_x^a(A) := \{\alpha \in \mathbb{R} : (a-x)\cdot y=…
The central place models are fundamentally important in theoretical geography and city planning theory. The texture and structure of central place networks have been demonstrated to be self-similar in both theoretical and empirical studies.…
Statistical learning in high-dimensional spaces is challenging without a strong underlying data structure. Recent advances with foundational models suggest that text and image data contain such hidden structures, which help mitigate the…
Measuring the complexity of high-dimensional data in physical systems becomes a critical factor in determining the information and quality of the systems. However, traditional metrics, such as Lyapunov exponent, fractal dimension, and…
In the absence of governing equations, dimensional analysis is a robust technique for extracting insights and finding symmetries in physical systems. Given measurement variables and parameters, the Buckingham Pi theorem provides a procedure…
Fractal dimension is widely adopted in spatial databases and data mining, among others as a measure of dataset skewness. State-of-the-art algorithms for estimating the fractal dimension exhibit linear runtime complexity whether based on…
The spatial distribution of unvisited/persistent sites in $d=1$ $A+A\to\emptyset$ model is studied numerically. Over length scales smaller than a cut-off $\xi(t)\sim t^{z}$, the set of unvisited sites is found to be a fractal. The fractal…
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…
Fractures are normally present in the underground and are, for some physical processes, of paramount importance. Their accurate description is fundamental to obtain reliable numerical outcomes useful, e.g., for energy management. Depending…
The image fractal analysis is actively used in all science branches. In particular in materials science the fractal analysis is applied to study microstructure of deformed metals because its structure can be interpreted as the fractal…
Diffusion Limited Aggregation (DLA) is a model of fractal growth that was introduced in 1981 and had since attained a paradigmatic status due to its simplicity and its underlying role for a variety of pattern forming processes. Despite…
Dimensionality reduction is a common method for analyzing and visualizing high-dimensional data across domains. Dimensionality-reduction algorithms involve complex optimizations and the reduced dimensions computed by these algorithms…
For better learning, large datasets are often split into small batches and fed sequentially to the predictive model. In this paper, we study such batch decompositions from a probabilistic perspective. We assume that data points (possibly…
Most biological data are multidimensional, posing a major challenge to human comprehension and computational analysis. Principal component analysis is the most popular approach to rendering two- or three-dimensional representations of the…
This paper discusses the critical decision process of extracting or selecting the features in a supervised learning context. It is often confusing to find a suitable method to reduce dimensionality. There are pros and cons to deciding…