Related papers: Joint Geometric and Topological Analysis of Hierar…

We use topological data analysis (TDA) to study how data transforms as it passes through successive layers of a deep neural network (DNN). We compute the persistent homology of the activation data for each layer of the network and summarize…

Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional…

The study of irreducible higher-order interactions has become a core topic of study in complex systems. Two of the most well-developed frameworks, topological data analysis and multivariate information theory, aim to provide formal tools…

In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single filtered space often does not adequately…

Topological data analysis (TDA) is a rising branch in modern applied mathematics. It extracts topological structures as features of a given space and uses these features to analyze digital data. Persistent homology, one of the central tools…

This paper introduces and demonstrates a computational pipeline for the statistical analysis of shape graph datasets, namely geometric networks embedded in 2D or 3D spaces. Unlike traditional abstract graphs, our purpose is not only to…

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…

An ultrametric topology formalizes the notion of hierarchical structure. An ultrametric embedding, referred to here as ultrametricity, is implied by a hierarchical embedding. Such hierarchical structure can be global in the data set, or…

Topological Data Analysis (TDA) has emerged as a powerful framework for extracting robust and interpretable features from noisy high-dimensional data. In the context of Social Choice Theory, where preference profiles and collective…

This study presents a hierarchical mining framework for high-dimensional imbalanced data, leveraging a depth graph model to address the inherent performance limitations of conventional approaches in handling complex, high-dimensional data…

Deep generative models have emerged as a powerful tool for learning useful molecular representations and designing novel molecules with desired properties, with applications in drug discovery and material design. However, most existing deep…

Identifying molecular signatures from complex disease patients with underlying symptomatic similarities is a significant challenge in the analysis of high dimensional multi-omics data. Topological data analysis (TDA) provides a way of…

Appropriately representing elements in a database so that queries may be accurately matched is a central task in information retrieval; recently, this has been achieved by embedding the graphical structure of the database into a manifold in…

Advances in imaging techniques enable high resolution 3D visualisation of vascular networks over time and reveal abnormal structural features such as twists and loops, and their quantification is an active area of research. Here we showcase…

Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. "Structure" can be understood as symmetry and a range of symmetries are expressed by hierarchy. Such symmetries directly…

Diversity can be broadly defined as the presence of meaningful variation across elements, which can be viewed from multiple perspectives, including statistical variation and geometric structural richness in the dataset. Existing diversity…

In this study, we present and analyze a framework for geometric and topological estimation for mapping of unknown environments. We consider agents mimicking motion behaviors of cyborg insects, known as biobots, and exploit coordinate-free…

Topological Data Analysis (TDA) is a recent approach to analyze data sets from the perspective of their topological structure. Its use for time series data has been limited. In this work, a system developed for a leading provider of cloud…

Outlier or anomaly detection is an important task in data analysis. We discuss the problem from a geometrical perspective and provide a framework that exploits the metric structure of a data set. Our approach rests on the manifold…

Topological Data Analysis (TDA), an emerging field in investment sciences, harnesses mathematical methods to extract data features based on shape, offering a promising alternative to classical portfolio selection methodologies. We utilize…