Related papers: A persistence landscapes toolbox for topological s…
Topological data analysis is becoming a popular way to study high dimensional feature spaces without any contextual clues or assumptions. This paper concerns itself with one popular topological feature, which is the number of…
We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of level sets of both density and regression functions. Our method is based on kernel estimation. We apply this…
Garside et al. use event history methods to analyze topological data. We provide additional background on persistent homology to contrast the hazard estimators used by Garside et al. with traditional approaches in topological data analysis.…
Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a…
The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of…
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…
In recent years there has been noticeable interest in the study of the "shape of data". Among the many ways a "shape" could be defined, topology is the most general one, as it describes an object in terms of its connectivity structure:…
Random walks on multidimensional nonlinear landscapes are of interest in many areas of science and engineering. In particular, properties of adaptive trajectories on fitness landscapes determine population fates and thus play a central role…
High order networks are weighted hypergraphs col- lecting relationships between elements of tuples, not necessarily pairs. Valid metric distances between high order networks have been defined but they are difficult to compute when the…
The theory of persistence modules is an emerging field of algebraic topology which originated in topological data analysis. In these notes we provide a concise introduction into this field and give an account on some of its interactions…
Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, persistence diagrams…
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…
Persistent homology is a powerful tool for characterizing the topology of a data set at various geometric scales. When applied to the description of molecular structures, persistent homology can capture the multiscale geometric features and…
In machine learning, a loss function measures the difference between model predictions and ground-truth (or target) values. For neural network models, visualizing how this loss changes as model parameters are varied can provide insights…
Information networks are becoming increasingly popular to capture complex relationships across various disciplines, such as social networks, citation networks, and biological networks. The primary challenge in this domain is measuring…
One of the most common problem-solving heuristics is by analogy. For a given problem, a solver can be viewed as a strategic walk on its fitness landscape. Thus if a solver works for one problem instance, we expect it will also be effective…
Topological data analysis has emerged as a powerful tool for extracting the metric, geometric and topological features underlying the data as a multi-resolution summary statistic, and has found applications in several areas where data…
This paper establishes connections between three of the most prominent metrics used in the analysis of persistence diagrams in topological data analysis: the bottleneck distance, Patel's erosion distance, and Bubenik's landscape distance.…
Topological data analysis (TDA) is a rising field in the intersection of mathematics, statistics, and computer science/data science. The cornerstone of TDA is persistent homology, which produces a summary of topological information called a…
In urgent decision making applications, ensemble simulations are an important way to determine different outcome scenarios based on currently available data. In this paper, we will analyze the output of ensemble simulations by considering…