Related papers: Classification of Firn Data via Topological Featur…
Topological Data Analysis (TDA) has been successfully used for various tasks in signal/image processing, from visualization to supervised/unsupervised classification. Often, topological characteristics are obtained from persistent homology…
We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems. Topological data analysis provides a set of tools to characterise the shape of data, in terms of…
Spectral clustering has gained importance in recent years due to its ability to cluster complex data as it requires only pairwise similarity among data points with its ease of implementation. The central point in spectral clustering is the…
Modern business and economic datasets often exhibit nonlinear, multi-scale structures that traditional linear tools under-represent. Topological Data Analysis (TDA) offers a geometric lens for uncovering robust patterns, such as connected…
3D point clouds of natural environments relevant to problems in geomorphology often require classification of the data into elementary relevant classes. A typical example is the separation of riparian vegetation from ground in fluvial…
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,…
Modern deep neural networks have shown remarkable performance in medical image classification. However, such networks either emphasize pixel-intensity features instead of fundamental anatomical structures (e.g., those encoded by topological…
Artificial Neural Networks (ANNs) require significant amounts of data and computational resources to achieve high effectiveness in performing the tasks for which they are trained. To reduce resource demands, various techniques, such as…
Unsupervised feature learning often finds low-dimensional embeddings that capture the structure of complex data. For tasks for which prior expert topological knowledge is available, incorporating this into the learned representation may…
We introduce a very general approach to the analysis of signals from their noisy measurements from the perspective of Topological Data Analysis (TDA). While TDA has emerged as a powerful analytical tool for data with pronounced topological…
We perform topological data analysis on the internal states of convolutional deep neural networks to develop an understanding of the computations that they perform. We apply this understanding to modify the computations so as to (a) speed…
A clustering algorithm partitions a set of data points into smaller sets (clusters) such that each subset is more tightly packed than the whole. Many approaches to clustering translate the vector data into a graph with edges reflecting a…
The field of face recognition (FR) has undergone significant advancements with the rise of deep learning. Recently, the success of unsupervised learning and graph neural networks has demonstrated the effectiveness of data structure…
Topological Data Analysis (TDA) is a rising field of computational topology in which the topological structure of a data set can be observed by persistent homology. By considering a sequence of sublevel sets, one obtains a filtration that…
Complex prediction models such as deep learning are the output from fitting machine learning, neural networks, or AI models to a set of training data. These are now standard tools in science. A key challenge with the current generation of…
Understanding the structure of Earth's polar ice sheets is important for modeling how global warming will impact polar ice and, in turn, the Earth's climate. Ground-penetrating radar is able to collect observations of the internal structure…
Topological Data Analysis (TDA) can be used to detect and characterize holes in an image, such as zero-dimensional holes (connected components) or one-dimensional holes (loops). However, there is currently no widely accepted statistical…
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 main focus of this work is on providing a formal definition of statistical depth for functional data on the basis of six properties, recognising topological features such as continuity, smoothness and contiguity. Amongst our depth…
We specialize techniques from topological data analysis to the problem of characterizing the topological complexity (as defined in the body of the paper) of a multi-class data set. As a by-product, a topological classifier is defined that…