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Meaningful topological invariants for mixed quantum states are challenging to identify as there is no unique way to define them, and most choices do not directly relate to physical observables. Here, we propose a simple pragmatic approach…

Topological data analysis is an approach to study shape of a data set by means of topology. Its main object of study is the persistence diagram, which represents the topological features of the data set at different spatial resolutions.…

The use of Variational Autoencoders in different Machine Learning tasks has drastically increased in the last years. They have been developed as denoising, clustering and generative tools, highlighting a large potential in a wide range of…

Topological structures in image data, such as connected components and loops, play a crucial role in understanding image content (e.g., biomedical objects). % Despite remarkable successes of numerous image processing methods that rely on…

We present a new technique that enables manifold learning to accurately embed data manifolds that contain holes, without discarding any topological information. Manifold learning aims to embed high dimensional data into a lower dimensional…

Topological data analysis is a powerful framework for extracting useful topological information from complex datasets. Recent work has shown its application for the dynamical analysis of classical dissipative systems through a…

Despite significant advances in the field of deep learning in applications to various fields, explaining the inner processes of deep learning models remains an important and open question. The purpose of this article is to describe and…

Owing to the advances in computational techniques and the increase in computational power, atomistic simulations of materials can simulate large systems with higher accuracy. Complex phenomena can be observed in such state-of-the-art…

As datasets used in scientific applications become more complex, studying the geometry and topology of data has become an increasingly prevalent part of the data analysis process. This can be seen for example with the growing interest in…

Topological Data Analysis (TDA) studies the shape of data. A common topological descriptor is the persistence diagram, which encodes topological features in a topological space at different scales. Turner, Mukeherjee, and Boyer showed that…

In this paper, we develop topological data analysis methods for classification tasks on univariate time series. As an application, we perform binary and ternary classification tasks on two public datasets that consist of physiological…

Topological data analysis (TDA) is a rapidly evolving field in applied mathematics and data science that leverages tools from topology to uncover robust, shape-driven insights in complex datasets. The main workhorse is persistent homology,…

Persistence diagrams have been widely used to quantify the underlying features of filtered topological spaces in data visualization. In many applications, computing distances between diagrams is essential; however, computing these distances…

Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data --- for example, to approximate a point cloud by a low-dimensional…

The discovery of topological phases in condensed matter systems has changed the modern conception of phases of matter. The global nature of topological ordering makes these phases robust and hence promising for applications. However, the…

Medical image segmentation, which aims to automatically extract anatomical or pathological structures, plays a key role in computer-aided diagnosis and disease analysis. Despite the problem has been widely studied, existing methods are…

Supervised manifold learning methods learn data representations by preserving the geometric structure of data while enhancing the separation between data samples from different classes. In this work, we propose a theoretical study of…

Hyperuniformity, the suppression of density fluctuations at large length scales, is observed across a wide variety of domains, from cosmology to condensed matter and biological systems. Although the standard definition of hyperuniformity…

We propose a novel approach for preserving topological structures of the input space in latent representations of autoencoders. Using persistent homology, a technique from topological data analysis, we calculate topological signatures of…

In this work we use the persistent homology method, a technique in topological data analysis (TDA), to extract essential topological features from the data space and combine them with deep learning features for classification tasks. In TDA,…