Related papers: Computing Persistent Homology within Coq/SSReflect
The present contribution investigates multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. Existing limit theorems for topological statistics prove difficult to use in…
The persistence diagram is a central object in the study of persistent homology and has also been investigated in the context of random topology. The more recent notion of the verbose diagram (a.k.a. verbose barcode) is a refinement of the…
Topological data analysis (TDA) has emerged as an effective approach in data science, with its key technique, persistent homology, rooted in algebraic topology. Although alternative approaches based on differential topology, geometric…
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…
Topological Data Analysis (TDA) offers a suite of computational tools that provide quantified shape features in high dimensional data that can be used by modern statistical and predictive machine learning (ML) models. In particular,…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
Recent years have witnessed a fast growth in mathematical artificial intelligence (AI). One of the most successful mathematical AI approaches is topological data analysis (TDA) via persistent homology (PH) that provides explainable AI (xAI)…
Persistent homology provides information about the lifetime of homology classes along a filtration of cell complexes. Persistence barcode is a graphical representation of such information. A filtration might be determined by time in a set…
We give an algorithm with singly exponential complexity for computing the barcodes up to dimension $\ell$ (for any fixed $\ell \geq 0$) of the filtration of a given semi-algebraic set by the sub-level sets of a given polynomial. Our…
In the realm of formal theorem proving, the Coq proof assistant stands out for its rigorous approach to verifying mathematical assertions and software correctness. Despite the advances in artificial intelligence and machine learning, the…
A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent…
Persistence diagrams, which summarize the birth and death of homological features extracted from data, are employed as stable signatures for applications in image analysis and other areas. Besides simply considering the multiset of…
Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jablonski,…
This is a survey article on the stable cohomotopy refinement of Seiberg-Witten invariants containing also new results, for example: - Stable cohomotopy groups describe path components of certain mapping spaces. - Relation of stable…
Topologically ordered phases of matter display a number of unique characteristics, including ground states that can be interpreted as patterns of closed strings. In this paper, we consider the problem of detecting and distinguishing closed…
The Extended Persistent Homology Transform (XPHT) is a topological transform which takes as input a shape embedded in Euclidean space, and to each unit vector assigns the extended persistence module of the height function over that shape…
Proof assistants like Coq are increasingly popular to help mathematicians carry out proofs of the results they conjecture. However, formal proofs remain highly technical and are especially difficult to reuse. In this paper, we present a…
Topology is the foundation for many industrial applications ranging from CAD to simulation analysis. Computational topology mostly focuses on structured data such as mesh, however unstructured dataset such as point set remains a virgin land…
The concept of topological persistence, introduced recently in computational topology, finds applications in studying a map in relation to the topology of its domain. Since its introduction, it has been extended and generalized in various…
The objective of this study is to detect and quantify the periodic behavior of the signals using topological methods. We propose to use delay-coordinate embeddings as a tool to measure the periodicity of signals. Moreover, we use persistent…