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Statistical topological data analysis using persistence landscapes
TLDR
A new topological summary for data that is easy to combine with tools from statistics and machine learning and obeys a strong law of large numbers and a central limit theorem is defined.
Categorification of Persistent Homology
TLDR
This work redevelops persistent homology (topological persistence) from a categorical point of view and gives a natural construction of a category of ε-interleavings of $\mathbf {(\mathbb {R},\leq)}$-indexed diagrams in some target category and shows that if the target category is abelian, so is this category of interleavments.
Metrics for Generalized Persistence Modules
TLDR
This work considers the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets, and introduces a distinction between ‘soft’ and ‘hard’ stability theorems.
Statistical topology using persistence landscapes
TLDR
A new descriptor for persistent homology is defined, which is thought of as an embedding of the usual descriptors, barcodes and persistence diagrams, into a space of functions, which inherits an L norm, and it is shown that this metric space is complete and separable.
Persistence Diagrams of Cortical Surface Data
TLDR
This work presents a novel framework for characterizing signals in images using techniques from computational algebraic topology, which uses all the local critical values in characterizing the signal and offers a completely new data reduction and analysis framework for quantifying the signal.
A statistical approach to persistent homology
TLDR
Using statistical estimators for samples from certain families of distributions, it is shown that the persistent homology of the underlying distribution can be recovered.
Persistent homology detects curvature
TLDR
It is proved that persistent homology detects the curvature of disks from which points have been sampled, and a general computational framework for solving inverse problems using the average persistence landscape, a continuous mapping from metric spaces with a probability measure to a Hilbert space is described.
Topological spaces of persistence modules and their properties
TLDR
This work considers various classes of persistence modules, including many of those that have been previously studied, and describes the relationships between them, and undertake a systematic study of the resulting topological spaces and their basic topological properties.
Embeddings of Persistence Diagrams into Hilbert Spaces
TLDR
This work shows that any separable, bounded metric space isometrically embeds into the space of persistence diagrams with the bottleneck distance, and obtains the generalized roundness, negative type, and asymptotic dimension of this space.
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