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Proximity of persistence modules and their diagrams
TLDR
This paper presents new stability results that do not suffer from the restrictions of existing stability results, and makes it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence.
The Structure and Stability of Persistence Modules
This book is a comprehensive treatment of the theory of persistence modules over the real line. It presents a set of mathematical tools to analyse the structure and to establish the stability of such
Persistence stability for geometric complexes
TLDR
The properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, Čech and witness complexes) built on top of totally bounded metric spaces are studied.
Towards persistence-based reconstruction in euclidean spaces
TLDR
A novel approach that stands in-between classical reconstruction and topological estimation, and whose complexity scales up with the intrinsic dimension of the data is introduced.
Persistence-Based Clustering in Riemannian Manifolds
TLDR
A clustering scheme that combines a mode-seeking phase with a cluster merging phase in the corresponding density map, and whose output clusters have the property that their spatial locations are bound to the ones of the basins of attraction of the peaks of the density.
Geometric Inference for Probability Measures
TLDR
Replacing compact subsets by measures, a notion of distance function to a probability distribution in ℝd is introduced and it is shown that it is possible to reconstruct offsets of sampled shapes with topological guarantees even in the presence of outliers.
An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists
TLDR
This paper is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of TDA for non experts.
Robust Topological Inference: Distance To a Measure and Kernel Distance
TLDR
The distance-to-a-measure (DTM), and the kernel distance, introduced by Phillips et al. (2014), are smooth functions that provide useful topological information but are robust to noise and outliers.
PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures
TLDR
This work shows how graphs can be encoded by (extended) persistence diagrams in a provably stable way and proposes a general and versatile framework for learning vectorizations of persistence diagrams, which encompasses most of the vectorization techniques used in the literature.
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