International audience
Given a continuous function $f:X\to\mathbb{R}$ and a cover $\mathcal{I}$ of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover $f^{-1}(\mathcal{I})$. Despite its success in applications, little is known about the structure and stability of this construction from a theoretical point of view. As a pixelized...
As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for applications, where it matters to quantify the extent by which two given Reeb graphs differ. Recent...
Given a continuous function f : X → R and a cover I of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover f −1 (I). Despite its success in applications, little is known about the structure and stability of this construction from a theoretical point of view. As a pixelized version of the Reeb graph of f , it is...
In the last decade, there has been increasing interest in topological data analysis, a new methodology for using geometric structures in data for inference and learning. A central theme in the area is the idea of persistence, which in its most basic form studies how measures of shape change as a scale parameter varies. There are now a number of...
Topological Data Analysis is a growing area of data science, which aims at computing and characterizing the geometry and topology of data sets, in order to produce useful descriptors for subsequent statistical and machine learning tasks. Its main computational tool is persistent homology, which amounts to track the topological changes in...
Topological Data Analysis (TDA) provides a pipeline to extract quantitative topological descriptors from structured objects. This enables the definition of topological loss functions, which assert to what extent a given object exhibits some topological properties. These losses can then be used to perform topological optimization via gradient...
In this article, we address the problem of devising signatures using the framework of persistent homology.Considering a compact length space with curvature bounded above, we build, either for every point or for the shape itself, a topological signature that is provably stable to perturbations of the space in the Gromov-Hausdorff distance. This...
In this article, we study the question of the statistical convergence of the 1-dimensional Mapper to its continuous analogue, the Reeb graph. We show that the Mapper is an optimal estimator of the Reeb graph, which gives, as a byproduct, a method to automatically tune its parameters and compute confidence regions on its topological features,...
Comparing points on 3D shapes is among the fundamental operations in shape analysis. To facilitate this task, a great number of local point signatures or descriptors have been proposed in the past decades. However, the vast majority of these descriptors concentrate on the local geometry of the shape around the point, and thus are insensitive to...
Topological data analysis (TDA) is a rapidly growing area of data science which uses the geometry and topology of data sets to produce qualitative multi-scale shape descriptors for subsequent statistical and machine learning tasks. The most common descriptor in TDA is persistent homology, which tracks the topological changes in growing families...
The generalized persistence diagram (GPD) is a natural extension of the classical persistence barcode to the setting of multi-parameter persistence and beyond. The GPD is defined as an integer-valued function whose domain is the set of intervals in the indexing poset of a persistence module, and is known to be able to capture richer topological...
Persistence diagrams (PDs) play a key role in topological data analysis (TDA), in which they are routinely used to describe topological properties of complicated shapes. PDs enjoy strong stability properties and have proven their utility in various learning contexts. They do not, however , live in a space naturally endowed with a Hilbert...
We introduce algorithms for robustly computing intrinsic coordinates on point clouds. Our approach relies on generating many candidate coordinates by subsampling the data and varying hyperparameters of the embedding algorithm (e.g., manifold learning). We then identify a subset of representative embeddings by clustering the collection of...
BackgroundThis paper exploits recent developments in topological data analysis to present a pipeline for clustering based on Mapper, an algorithm that reduces complex data into a one-dimensional graph.ResultsWe present a pipeline to identify and summarise clusters based on statistically significant topological features from a point cloud using...
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-convexoptimization, statistics and machine learning. However, the approaches proposed in the literatureare usually anchored to a specific...
Precision medicine allows the extraction of information from complex datasets to facilitate clinical decision-making at the individual level. Topological Data Analysis (TDA) offers promising tools that complement current analytical methods in precision medicine studies. We introduce the fundamental concepts of the TDA corpus (the simplicial...
Persistent homology is a powerful tool in topological data analysis. The main output, persistence diagrams, encode the geometry and topology of given datasets. We present a novel application of persistent homology to characterize the biological environment surrounding breast cancers, known as the tumor microenvironment. Specifically, we will...
Graph classification is a difficult problem that has drawn a lot of attention from the machine learning community over the past few years. This is mainly due to the fact that, contrarily to Euclidean vectors, the inherent complexity of graph structures can be quite hard to encode and handle for traditional classifiers. Even though kernels have...