Related papers: Persistence Spheres: a Bi-continuous Linear Repres…
We introduce persistence spheres, a novel functional representation of persistence diagrams. Unlike existing embeddings (such as persistence images, landscapes, or kernel methods), persistence spheres provide a bi-continuous mapping: they…
Despite the obvious similarities between the metrics used in topological data analysis and those of optimal transport, an optimal-transport based formalism to study persistence diagrams and similar topological descriptors has yet to come.…
Topological methods for data analysis present opportunities for enforcing certain invariances of broad interest in computer vision, including view-point in activity analysis, articulation in shape analysis, and measurement invariance in…
Many attempts have been made in recent decades to integrate machine learning (ML) and topological data analysis. A prominent problem in applying persistent homology to ML tasks is finding a vector representation of a persistence diagram…
Segmenting curvilinear structures in medical images is essential for analyzing morphological patterns in clinical applications. Integrating topological properties, such as connectivity, improves segmentation accuracy and consistency.…
Modern representation learning increasingly relies on unsupervised and self-supervised methods trained on large-scale unlabeled data. While these approaches achieve impressive generalization across tasks and domains, evaluating embedding…
We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning…
Persistence diagrams, an important summary in topological data analysis, consist of a set of ordered pairs, each with positive multiplicity. Persistence diagrams are obtained via Mobius inversion and may be compared using a one-parameter…
This article addresses regularity of optimal transport maps for cost="squared distance" on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be…
Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a…
Persistence-based topological optimization deforms a point cloud $X \subset \mathbb{R}^d$ by minimizing objectives of the form $L(X) = \ell(\mathrm{Dgm}(X))$, where $\mathrm{Dgm}(X)$ is a persistence diagram. In practice, optimization is…
Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of complex data. Despite several appealing properties, incorporating PDs in learning pipelines can be challenging because their natural geometry is not…
Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane $\mathbb{R}^2$ that can…
Persistence diagrams are common descriptors of the topological structure of data appearing in various classification and regression tasks. They can be generalized to Radon measures supported on the birth-death plane and endowed with an…
Building upon [2308.02636], we investigate the constraining power of persistent homology on cosmological parameters and primordial non-Gaussianity in a likelihood-free inference pipeline utilizing machine learning. We evaluate the ability…
As the most common representation for 3D shapes, mesh is often stored discretely with arrays of vertices and faces. However, 3D shapes in the real world are presented continuously. In this paper, we propose to learn a continuous…
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
Topological data analysis (TDA) is a rapidly developing collection of methods for studying the shape of point cloud and other data types. One popular approach, designed to be robust to noise and outliers, is to first use a smoothing…
The contribution of this work is twofold. The first part deals with a Hilbert-space version of McCann's celebrated result on the existence and uniqueness of monotone measure-preserving maps: given two probability measures $\rm P$ and $\rm…
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…