Related papers: The density of expected persistence diagrams and i…
Topological data analysis provides a multiscale description of the geometry and topology of quantitative data. The persistence landscape is a topological summary that can be easily combined with tools from statistics and machine learning.…
We introduce \emph{topological density estimation} (TDE), in which the multimodal structure of a probability density function is topologically inferred and subsequently used to perform bandwidth selection for kernel density estimation. We…
In this work, we develop a pipeline that associates Persistence Diagrams to digital data via the most appropriate filtration for the type of data considered. Using a grid search approach, this pipeline determines optimal representation…
We introduce a linear dimensionality reduction technique preserving topological features via persistent homology. The method is designed to find linear projection $L$ which preserves the persistent diagram of a point cloud $\mathbb{X}$ via…
The microstructure critically governs the properties of materials used in energy and chemical engineering technologies, from catalysts and filters to thermal insulators and sensors. Therefore, accurate design is based on quantitative…
Persistence diagrams are important tools in the field of topological data analysis that describe the presence and magnitude of features in a filtered topological space. However, current approaches for comparing a persistence diagram to a…
Many predictions are probabilistic in nature; for example, a prediction could be for precipitation tomorrow, but with only a 30 percent chance. Given both the predictions and the actual outcomes, "reliability diagrams" (also known as…
In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…
After extensive quasi-static shearing, dense dry granular flows attain a steady-state condition of porosity and deviatoric stress, even as particles are continually rearranged. The paper considers two-dimensional flow and derives the…
Graph-based representations of point-cloud data are widely used in data science and machine learning, including epsilon-graphs that contain edges between pairs of data points that are nearer than epsilon and kNN-graphs that connect each…
While spherical data arises in many contexts, including in directional statistics, the current tools for density estimation and population comparison on spheres are quite limited. Popular approaches for comparing populations (on Euclidean…
Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of…
In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on…
Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a…
Persistence diagrams (PDs), often characterized as sets of death and birth of homology class, have been known for providing a topological representation of a graph structure, which is often useful in machine learning tasks. Prior works rely…
We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered…
Neural Persistence is a prominent measure for quantifying neural network complexity, proposed in the emerging field of topological data analysis in deep learning. In this work, however, we find both theoretically and empirically that the…
A method is presented for the distributed computation of persistent homology, based on an extension of the generalized Mayer-Vietoris principle to filtered spaces. Cellular cosheaves and spectral sequences are used to compute global…
Computation of the simplicial complexes of a large point cloud often relies on extracting a sample, to reduce the associated computational burden. The study considers sampling critical points of a Morse function associated to a point cloud,…
Data analysis in high-dimensional spaces aims at obtaining a synthetic description of a data set, revealing its main structure and its salient features. We here introduce an approach providing this description in the form of a topography of…