# Persistence Flamelets: multiscale Persistent Homology for kernel density exploration

@article{Padellini2017PersistenceFM, title={Persistence Flamelets: multiscale Persistent Homology for kernel density exploration}, author={Tullia Padellini and Pierpaolo Brutti}, journal={arXiv: Machine Learning}, year={2017} }

In recent years there has been noticeable interest in the study of the "shape of data". Among the many ways a "shape" could be defined, topology is the most general one, as it describes an object in terms of its connectivity structure: connected components (topological features of dimension 0), cycles (features of dimension 1) and so on. There is a growing number of techniques, generally denoted as Topological Data Analysis, aimed at estimating topological invariants of a fixed object; when we…

## 4 Citations

### Approximating Continuous Functions on Persistence Diagrams Using Template Functions

- Computer ScienceFoundations of Computational Mathematics
- 2022

This paper describes a mathematical framework for featurizing the persistence diagram space using template functions, and discusses two example realizations of these functions: tent functions and Chybeyshev interpolating polynomials.

### Functional summaries of persistence diagrams

- Computer ScienceJ. Appl. Comput. Topol.
- 2020

The definition of persistence landscape functions is generalized, several theoretical properties of the persistence functional summaries are established, and their performance in the context of classification using simulated prostate cancer histology data is demonstrated.

### Persistent Homology of Complex Networks for Dynamic State Detection

- Computer SciencePhysical review. E
- 2019

It is shown how persistent homology, a tool from TDA, can be used to yield a compressed, multi-scale representation of the graph that can distinguish between dynamic states such as periodic and chaotic behavior.

### Supervised learning with indefinite topological Kernels

- Computer ScienceStatistics
- 2021

This work introduces a new exponential kernel, built on the geodesic space of PDs, and shows with simulated and real applications how it can be successfully used in regression and classification tasks, despite not being positive definite.

## References

SHOWING 1-10 OF 24 REFERENCES

### Stochastic Convergence of Persistence Landscapes and Silhouettes

- MathematicsJ. Comput. Geom.
- 2015

An alternate functional summary of persistent homology is introduced, which is called the silhouette, and an analogous statistical theory is derived that investigates the statistical properties of landscapes, such as weak convergence of the average landscapes and convergence ofThe bootstrap.

### Vines and vineyards by updating persistence in linear time

- Mathematics, Computer ScienceSCG '06
- 2006

The main result of this paper is an algorithm that maintains the pairing in worst-case linear time per transposition in the ordering and uses the algorithm to compute 1-parameter families of diagrams which are applied to the study of protein folding trajectories.

### Homological illusions of persistence and stability

- Mathematics
- 2008

In this thesis we explore and extend the theory of persistent homology, which captures topological features of a function by pairing its critical values. The result is represented by a collection of…

### Confidence sets for persistence diagrams

- MathematicsThe Annals of Statistics
- 2014

This paper derives confidence sets that allow us to separate topological signal from topological noise, and brings some statistical ideas to persistent homology.

### Statistical topological data analysis using persistence landscapes

- MathematicsJ. Mach. Learn. Res.
- 2015

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.

### Finding Singular Features

- Computer Science
- 2016

This work shows how to find singular features by first finding ridges in the estimated density, followed by a filtering step based on the eigenvalues of the Hessian of the density, which outputs well-defined sets of dimensions d < D.

### The Structure and Stability of Persistence Modules

- MathematicsSpringer Briefs in Mathematics
- 2016

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…

### Probability measures on the space of persistence diagrams

- Mathematics
- 2011

This paper shows that the space of persistence diagrams has properties that allow for the definition of probability measures which support expectations, variances, percentiles and conditional…

### Topology and data

- Computer Science
- 2009

This paper will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data, particularly high throughput data from microarray or other sources.

### Non‐parametric inference for density modes

- Computer ScienceArXiv
- 2013

This work derives non‐parametric confidence intervals for the eigenvalues of the Hessian at modes of a density estimate using a data‐splitting approach and suggests a new method for bandwidth selection, namely choosing the bandwidth to maximize the number of significant modes.