Related papers: Persistent Stanley--Reisner Theory
We apply persistent homology, the dominant tool from the field of topological data analysis, to study electoral redistricting. Our method combines the geographic information from a political districting plan with election data to produce a…
Persistent (co)homology is a central construction in topological data analysis, where it is used to quantify prominence of features in data to produce stable descriptors suitable for downstream analysis. Persistence is challenging to…
Persistent homology (PH) is one of the most popular tools in topological data analysis (TDA), while graph theory has had a significant impact on data science. Our earlier work introduced the persistent spectral graph (PSG) theory as a…
We propose a novel exact multi-parameter persistent homology method for analyzing time-series data utilizing the Liouville torus. In the field of topological data analysis (TDA), the conventional approach to analyzing time-series data often…
Unsupervised learning aims at the discovery of hidden structure that drives the observations in the real world. It is essential for success in modern machine learning. Latent variable models are versatile in unsupervised learning and have…
Current test-time adaptation (TTA) approaches aim to adapt a machine learning model to environments that change continuously. Yet, it is unclear whether TTA methods can maintain their adaptability over prolonged periods. To answer this…
Recently, persistent homology has had tremendous success in biomolecular data analysis. It works by examining the topological relationship or connectivity of a group of atoms in a molecule at a variety of scales, then rendering a family of…
Despite the notable advancements in numerous Transformer-based models, the task of long multi-horizon time series forecasting remains a persistent challenge, especially towards explainability. Focusing on commonly used saliency maps in…
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…
Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set…
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…
The study of phase transitions using data-driven approaches is challenging, especially when little prior knowledge of the system is available. Topological data analysis is an emerging framework for characterizing the shape of data and has…
Many modern datasets don't fit neatly into $n \times p$ matrices, but most techniques for measuring statistical stability expect rectangular data. We study methods for stability assessment on non-rectangular data, using statistical learning…
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…
This paper extends the key concept of persistence within Topological Data Analysis (TDA) in a new direction. TDA quantifies topological shapes hidden in unorganized data such as clouds of unordered points. In the 0-dimensional case the…
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…
While topological data analysis has emerged as a powerful paradigm for structural inference, its foundational tools, notably persistent homology and the persistent Laplacian, are frequently insensitive to localized structural fluctuations…
Multidimensional persistence modules do not admit a concise representation analogous to that provided by persistence diagrams for real-valued functions. However, there is no obstruction for multidimensional persistent Betti numbers to admit…
Given a data set with a notion of distance, such as a point cloud in Euclidean space, topological data analysis (TDA) uses techniques from algebraic topology and metric geometry to infer the topology of a hypothetical manifold from which…
Persistent homology computes topological invariants from point cloud data. Recent work has focused on developing statistical methods for data analysis in this framework. We show that, in certain models, parametric inference can be performed…