Related papers: Euler Characteristic Surfaces
Persistent topological properties of an image serve as an additional descriptor providing an insight that might not be discovered by traditional neural networks. The existing research in this area focuses primarily on efficiently…
Given a continuous sensor field, we can apply the Euler characteristic integral approach to count the number of targets in the sensor field. If the sensor field is discrete, the Euler integral approach introduces errors into our target…
Topological Data Analysis (TDA) is a novel statistical technique, particularly powerful for the analysis of large and high dimensional data sets. Much of TDA is based on the tool of persistent homology, represented visually via persistence…
Given a definable function $f: S \to \mathbb{R}$ on a definable set $S$, we study sublevel sets of the form $S^f_t \coloneqq \{x \in S: f(x) \leq t\}$ for all $t \in \mathbb{R}$. Using o-minimal structures, we prove that the Euler…
Shape recognition and classification is a problem with a wide variety of applications. Several recent works have demonstrated that topological descriptors can be used as summaries of shapes and utilized to compute distances. In this…
The higher characteristics w_m(G) for a finite abstract simplicial complex G are topological invariants that satisfy k-point Green function identities and can be computed in terms of Euler characteristic in the case of closed manifolds,…
Unsupervised data representation and visualization using tools from topology is an active and growing field of Topological Data Analysis (TDA) and data science. Its most prominent line of work is based on the so-called Mapper graph, which…
The aim of this paper is to derive on the basis of the Euler's formula several analytical relations which hold for certain classes of planar graphs and which can be useful in algorithmic graph theory.
Assume that the coefficients of a polynomial in a complex variable are Laurent polynomials in some complex parameters. The parameter space (a complex torus) splits into strata corresponding to different combinations of coincidence of the…
The Euler tour technique is a classical tool for designing parallel graph algorithms, originally proposed for the PRAM model. We ask whether it can be adapted to run efficiently on GPU. We focus on two established applications of the…
This paper is an introductory and informal exposition on the topology of polygonal meshes. We begin with a broad overview of topological notions and discuss how homeomorphisms, homotopy, and homology can be used to characterise topology. We…
This paper introduces advanced techniques of Topological Data Analysis (TDA) for unsupervised anomaly detection and customer segmentation in banking data. Using the Mapper algorithm and persistent homology, we develop unsupervised…
In recent years, topological data analysis has been utilized for a wide range of problems to deal with high dimensional noisy data. While text representations are often high dimensional and noisy, there are only a few work on the…
Topological data analysis is a relatively new branch of machine learning that excels in studying high dimensional data, and is theoretically known to be robust against noise. Meanwhile, data objects with mixed numeric and categorical…
We consider a topological integral transform of Bessel (concentric isospectral sets) type and Fourier (hyperplane isospectral sets) type, using the Euler characteristic as a measure. These transforms convert constructible $\zed$-valued…
The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data…
Understanding the topological characteristics of data is important to many areas of research. Recent work has demonstrated that synthetic 4D image-type data can be useful to train 4D convolutional neural network models to see topological…
Topological Data Analysis (TDA) has been praised by researchers for its ability to capture intricate shapes and structures within data. TDA is considered robust in handling noisy and high-dimensional datasets, and its interpretability is…
Statistical analysis on object data presents many challenges. Basic summaries such as means and variances are difficult to compute. We apply ideas from topology to study object data. We present a framework for using persistence landscapes…
The utilization of statistical methods an their applications within the new field of study known as Topological Data Analysis has has tremendous potential for broadening our exploration and understanding of complex, high-dimensional data…