# New families of simplicial filtration functors

@article{Chowdhury2017NewFO, title={New families of simplicial filtration functors}, author={Samir Chowdhury and Nathaniel Clause and Facundo M{\'e}moli and Jose Angel Sanchez and Zoe Wellner}, journal={arXiv: Algebraic Topology}, year={2017} }

The so called \v{C}ech and Vietoris-Rips simplicial filtrations are designed to capture information about the topological structure of metric datasets. These filtrations are two of the workhorses in the field of topological data analysis. They enjoy stability with respect to the Gromov-Hausdorff (GH) distance, and this stability property allows us to estimate the GH distance between finite metric space representations of the underlying datasets.
Via the concept of Gromov's curvature sets we…

## One Citation

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- Mathematics
- 2020

This thesis presents a uniform treatment of different distances used in the applied topology literature. We introduce the notion of a locally persistent category, which is a category with a notion of…

## References

SHOWING 1-10 OF 16 REFERENCES

### A Distance Between Filtered Spaces Via Tripods

- MathematicsArXiv
- 2017

A simplified treatment of stability of filtrations on finite spaces is presented and it is proved that this distance is intrinsic by constructing explicit geodesics between any pair of filtered spaces.

### Some Properties of Gromov–Hausdorff Distances

- MathematicsDiscret. Comput. Geom.
- 2012

A modified version of the Gromov–Hausdorff distance is studied which is motivated by practical applications and both prove a structural theorem for it and study its topological equivalence to the usual notion.

### Persistence barcodes for shapes

- Mathematics, Computer ScienceSGP '04
- 2004

This paper initiates a study of shape description and classification via the application of persistent homology to two tangential constructions on geometric objects, obtaining a shape descriptor, called a barcode, that is a finite union of intervals.

### Combining Persistent Homology and Invariance Groups for Shape Comparison

- MathematicsDiscret. Comput. Geom.
- 2016

A way to combine persistent homology with the use of G-invariant non-expansive operators defined on varPhi, where G is a group of self-homeomorphisms of X, is proposed.

### Characterization, Stability and Convergence of Hierarchical Clustering Methods

- MathematicsJ. Mach. Learn. Res.
- 2010

It is shown that within this framework, one can prove a theorem analogous to one of Kleinberg (2002), in which one obtains an existence and uniqueness theorem instead of a non-existence result.

### Improved Error Bounds for Tree Representations of Metric Spaces

- Computer ScienceNIPS
- 2016

This work considers an embedding of a metric space into a tree proposed by Gromov and obtains an improved additive distortion bound depending only on the hyperbolicity and doubling dimension of the metric.

### Measuring shapes by size functions

- Mathematics, Computer ScienceOther Conferences
- 1992

The concept of deformation distance between manifolds is presented, a distance which measures the `difference in shape' of two manifolds and the link between deformation distances and size functions is pointed out.

### Computational Topology - an Introduction

- Computer Science
- 2009

This book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles.

### Metric Structures for Riemannian and Non-Riemannian Spaces

- Mathematics
- 1999

Length Structures: Path Metric Spaces.- Degree and Dilatation.- Metric Structures on Families of Metric Spaces.- Convergence and Concentration of Metrics and Measures.- Loewner Rediscovered.-…

### Topological data analysis

- Mathematics
- 2011

Inverse problems can be defined as the area of mathematics that attempts to reconstruct a physical or mathematical object from derived data. Frequently, this means the evaluation of parameters or…