Related papers: Euler Characteristic Surfaces
Topological data analysis uses tools from topology -- the mathematical area that studies shapes -- to create representations of data. In particular, in persistent homology, one studies one-parameter families of spaces associated with data,…
In this paper we give an attempt to extend some arithmetic properties such as multiplicativity, convolution products to the setting of operators theory. We provide a significant examples which are of interest in number theory. We also give…
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
For the moduli spaces of Abelian differentials, the Euler characteristic is one of the most basic intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth…
In this paper we extend the notion of the Euler characteristic to persistent homology and give the relationship between the Euler integral of a function and the Euler characteristic of the function's persistent homology. We then proceed to…
Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discriminative than single measurements. The first two topological transforms to be defined…
The Mapper algorithm, a technique within topological data analysis (TDA), constructs a simplified graphical representation of high-dimensional data to uncover its underlying shape and structural patterns. The algorithm has attracted…
Multifields datasets are common in a large number of research and engineering applications of computational science. The effective visualization of the corresponding datasets can facilitate their analysis by elucidating the complex and…
Using a covariant description of the geometry of deformations for extendons, it is shown that the topological corrections for the string action associated with the Euler characteristic and the first Chern number of the normal bundle of the…
Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features,…
We prove the $L^2$-Euler characteristic has the invariance under the barycentric subdivision only for finite acyclic categories. And we extend the definition of $L^2$-Euler characteristic and prove the extended $L^2$-Euler characteristic…
With the growing adoption of AI-based systems across everyday life, the need to understand their decision-making mechanisms is correspondingly increasing. The level at which we can trust the statistical inferences made from AI-based…
We consider different generalizations of the Euler formula and discuss the properties of the associated trigonometric functions. The problem is analyzed from different points of view and it is shown that it can be formulated in a natural…
The Euler Characteristic Transform (ECT) is a robust method for shape classification. It takes an embedded shape and, for each direction, computes a piecewise constant function representing the Euler Characteristic of the shape's sublevel…
In a world abundant with diverse data arising from complex acquisition techniques, there is a growing need for new data analysis methods. In this paper we focus on high-dimensional data that are organized into several hierarchical datasets.…
We present an algorithm for the symbolic and numerical computation of the degrees of the Chern-Schwartz-MacPherson classes of a closed subvariety of projective space P^n. As the degree of the top Chern-Schwartz-MacPherson class is the…
Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such…
The identification of new rare signals in data, the detection of a sudden change in a trend, and the selection of competing models, are among the most challenging problems in statistical practice. These challenges can be tackled using a…
It is known to be difficult to find out whether a certain multivariable function to be a characteristic function when its corresponding measure is not tirivial to be or not to be a probability measure on R^d. Such results were not obtained…
We represent excursion sets of smooth random fields as unions of a topological basis consisting of a sequence of simply and multiply connected compact subsets of the underlying manifold. The associated coefficients, which are non-negative…