Related papers: Understanding the Power of Persistence Pairing via…
We present a new tool for data analysis: persistence discrete homology, which is well-suited to analyze filtrations of graphs. In particular, we provide a novel way of representing high-dimensional data as a filtration of graphs using…
The problem of detecting changes in covariance for a single pair of features has been studied in some detail, but may be limited in importance or general applicability. In contrast, testing equality of covariance matrices of a {\it set} of…
Topological Data Analysis (TDA) is a field that leverages tools and ideas from algebraic topology to provide robust methods for analysing geometric and topological aspects of data. One of the principal tools of TDA, persistent homology,…
Researchers often have datasets measuring features $x_{ij}$ of samples, such as test scores of students. In factor analysis and PCA, these features are thought to be influenced by unobserved factors, such as skills. Can we determine how…
We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric…
We consider finite-sample inference for a single regression coefficient in the fixed-design linear model $Y = Z\beta + bX + \varepsilon$, where $\varepsilon\in\mathbb{R}^n$ may exhibit complex dependence or heterogeneity. We develop a group…
Permutation-based partial-correlation tests guarantee finite-sample Type I error control under any fixed design and exchangeable noise, yet their power can collapse when the permutation-augmented design aligns too closely with the covariate…
Graph pattern matching is often defined in terms of subgraph isomorphism, an NP-complete problem. To lower its complexity, various extensions of graph simulation have been considered instead. These extensions allow pattern matching to be…
The machinery of topological data analysis becomes increasingly popular in a broad range of machine learning tasks, ranging from anomaly detection and manifold learning to graph classification. Persistent homology is one of the key…
The stability of persistent homology has led to wide applications of the persistence diagram as a trusted topological descriptor in the presence of noise. However, with the increasing demand for high-dimension and low-sample-size data…
This paper introduces a simple measure of a concordance pattern among observed outcomes along a network, i.e., the pattern in which adjacent outcomes tend to be more strongly correlated than non-adjacent outcomes. The graph concordance…
We consider the class of conditional graph patterns (\emph{CGPs}) that allow user to query data graphs with complex patterns that contain negation and predicates. To overcome the prohibitive cost of subgraph isomorphism, we consider…
Comparing two population means of network data is of paramount importance in a wide range of scientific applications. Many existing network inference solutions focus on global testing of entire networks, without comparing individual network…
Feature extraction from persistence diagrams, as a tool to enrich machine learning techniques, has received increasing attention in recent years. In this paper we explore an adaptive methodology to localize features in persistent diagrams,…
High-dimensional reduction methods are powerful tools for describing the main patterns in big data. One of these methods is the topological data analysis (TDA), which modeling the shape of the data in terms of topological properties. This…
Metric graphs are ubiquitous in science and engineering. For example, many data are drawn from hidden spaces that are graph-like, such as the cosmic web. A metric graph offers one of the simplest yet still meaningful ways to represent the…
Artificial neural networks can learn complex, salient data features to achieve a given task. On the opposite end of the spectrum, mathematically grounded methods such as topological data analysis allow users to design analysis pipelines…
Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations…
We introduce the persistence heatmap, a parametrized summary based on representative cycles in persistence diagrams, designed to enhance stability and explainability in topological data analysis. Algorithms to compute persistence diagrams…
Solving optimization tasks based on functions and losses with a topological flavor is a very active, growing field of research in data science and Topological Data Analysis, with applications in non-convex optimization, statistics and…