• Corpus ID: 8369823

Persistence weighted Gaussian kernel for topological data analysis

  title={Persistence weighted Gaussian kernel for topological data analysis},
  author={Genki Kusano and Yasuaki Hiraoka and Kenji Fukumizu},
Topological data analysis (TDA) is an emerging mathematical concept for characterizing shapes in complex data. In TDA, persistence diagrams are widely recognized as a useful descriptor of data, and can distinguish robust and noisy topological properties. This paper proposes a kernel method on persistence diagrams to develop a statistical framework in TDA. The proposed kernel satisfies the stability property and provides explicit control on the effect of persistence. Furthermore, the method… 
Kernel Method for Persistence Diagrams via Kernel Embedding and Weight Factor
A kernel method on persistence diagrams is proposed that allows one to control the effect of persistence, and, if necessary, noisy topological properties can be discounted in data analysis.
On the expectation of a persistence diagram by the persistence weighted kernel
  • G. Kusano
  • Mathematics, Computer Science
    Japan Journal of Industrial and Applied Mathematics
  • 2019
This paper studies relationships between a probability distribution and the persistence weighted kernel in the viewpoint of the strong law of large numbers and the central limit theorem, a confidence interval, and the stability theorem to ensure the continuity of the map from a probability Distribution to the expectation.
Persistence diagrams with linear machine learning models
A unified method for the inverse analysis is proposed by combining linear machine learning models with persistence images, applied to point clouds and cubical sets, showing the ability of the statistical inverse analysis and its advantages.
Persistence Weighted Gaussian Kernel for Probability Distributions on the Space of Persistence Diagrams
A persistence diagram characterizes robust geometric and topological features in data. Data, which will be treated here, are assumed to be drawn from a probability distribution and then the
Smooth Summaries of Persistence Diagrams and Texture Classification
The statistical properties of Gaussian persistence curves are investigated and applied to texture datasets: UIUCTex and KTH and the classification results on these texture datasets perform competitively with the current state-of-arts methods in TDA.
Supervised learning with indefinite topological Kernels
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.
Hypothesis Testing for Shapes using Vectorized Persistence Diagrams
This paper presents a two-stage hypothesis test for vectorized persistence diagrams, which shows that the proposed hypothesis tests can provide flexible and informative inferences on the shape of data with lower computational cost compared to the permutation test.
Persistence Curves: A canonical framework for summarizing persistence diagrams
This paper develops a general and unifying framework of vectorizing diagrams that it is shown that several well-known summaries, such as Persistence Landscapes, fall under the PC framework, and proposes several new summaries based on PC framework that provide a theoretical foundation for their stability analysis.
PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures
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.


Convergence rates for persistence diagram estimation in topological data analysis
It is shown that the use of persistent homology can be naturally considered in general statistical frameworks and established convergence rates of persistence diagrams associated to data randomly sampled from any compact metric space to a well defined limit diagram encoding the topological features of the support of the measure from which the data have been sampled.
Statistical Topological Data Analysis - A Kernel Perspective
This work proves universality of a variant of the original kernel on persistence diagrams, and demonstrates its effective use in two-sample hypothesis testing on synthetic as well as real-world data.
Statistical topological data analysis using persistence landscapes
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.
Confidence sets for persistence diagrams
This paper derives confidence sets that allow us to separate topological signal from topological noise, and brings some statistical ideas to persistent homology.
A stable multi-scale kernel for topological machine learning
This work designs a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data that is positive definite and proves its stability with respect to the 1-Wasserstein distance.
Persistence Images: An Alternative Persistent Homology Representation
It is shown that several machine learning techniques, applied to persistence images for classification tasks, yield high accuracy rates on multiple data sets and these sameMachine learning techniques fare better when applied to persistency images than when applied when it comes to persistence diagrams.
Hierarchical structures of amorphous solids characterized by persistent homology
A unified method using persistence diagrams for studying the geometry of atomic configurations in amorphous solids, which suggests that the PDs provide a unified method that extracts greater depth of geometric information in amortized solids than conventional methods.
Topological Measurement of Protein Compressibility via Persistence Diagrams
We exploit recent advances in computational topology to study the compressibility of various proteins found in the Protein Data Bank (PDB). Our fundamental tool is the persistence diagram, a
Persistent homology analysis of protein structure, flexibility, and folding
  • Kelin Xia, G. Wei
  • Biology
    International journal for numerical methods in biomedical engineering
  • 2014
The topology–function relationship of proteins is revealed, for the first time, and an excellent consistence between the persistent homology prediction and molecular dynamics simulation is found.