# Persistence Images: A Stable Vector Representation of Persistent Homology

@article{Adams2017PersistenceIA, title={Persistence Images: A Stable Vector Representation of Persistent Homology}, author={Henry Adams and Tegan H. Emerson and Michael J. Kirby and Rachel Neville and Chris Peterson and Patrick D. Shipman and Sofya Chepushtanova and Eric M. Hanson and Francis C. Motta and Lori Ziegelmeier}, journal={J. Mach. Learn. Res.}, year={2017}, volume={18}, pages={8:1-8:35} }

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 multiscale description of the homological features within a dataset. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning…

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## 336 Citations

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A new finite-dimensional vector is proposed, called the interconnectivity vector, representation of a PD adapted from Bag-of-Words (BoW), constructed to demonstrate the connections between the homological features of a data set.

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Persistence diagrams, a key descriptor from Topological Data Analysis, encode and summarize all sorts of topological features and have already proved pivotal in many different applications of data…

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This work proposes a new set of metrics based on persistence curves, and proves the stability of the proposed metrics, and applies these metrics to the UCR Time Series Classification Archive.

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

A general technique for extracting a larger set of stable information from persistent homology computations than is currently done by recast these problems as real-valued functions which are discontinuous but measurable, and observe that convolving a function with a suitable function produces a Lipschitz function.

Learning Hyperbolic Representations of Topological Features

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A method to learn representations of persistence diagrams on hyperbolic spaces, more specifically on the Poincare ball, which represents features of infinite persistence infinitesimally close to the boundary of the ball so their distance to non-essential features approaches infinity, thereby their relative importance is preserved.

Pseudo-Multidimensional Persistence and Its Applications

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- 2018

This new approach is able to differentiate between topologically equivalent geometric objects and offers insight into the study of the Kuramoto–Sivashinsky partial differential equation and Lissajous knots.

A Hybrid Metric based on Persistent Homology and its Application to Signal Classification

- Computer Science2020 25th International Conference on Pattern Recognition (ICPR)
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The main contributions of this work are to provide new sets of features for time series, prove that these features are robust to noise, and propose a hybrid metric that takes both geometric and topological information of the time series into account.

A computationally efficient framework for vector representation of persistence diagrams

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This work proposes a computationally efficient framework to convert a PD into a vector in R, called a vectorized persistence block (VPB), and shows that the representation possesses many of the desired properties of vector-based summaries such as stability with respect to input noise, low computational cost and flexibility.

Topological Machine Learning with Persistence Indicator Functions

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- 2019

This paper presents persistence indicator functions (PIFs), which summarize persistence diagrams, i.e., feature descriptors in topological data analysis, and demonstrates their usage in common data analysis scenarios, such as confidence set estimation and classification of complex structured data.

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