Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their {\em diagram distances}, most of the recent attempts at using persistence diagrams in machine learning have been done through kernel methods, i.e., embeddings of …
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48 results for “Persistence Diagram”
Develops robust persistence diagrams using kernel methods.
problem Persistence diagrams are sensitive to data perturbations.
method Constructs robust persistence diagrams from superlevel filtrations of robust density estimators using reproducing kernels.
result Robust persistence diagrams are consistent estimators in bottleneck distance.
Paper proves -means clustering works on persistence diagrams.
problem Complex geometry of persistence diagram space.
method Proves convergence of -means on persistence diagram space.
result Performance of -means on persistence diagrams and measures is superior.
Despite the obvious similarities between the metrics used in topological data analysis and those of optimal transport, an optimal-transport based formalism to study persistence diagrams and similar topological descriptors has yet to come. In this article, by considering the space of persistence diagrams as a space of d…
Revises SWK for persistence diagrams using Figalli-Gigli distance.
problem Efficiently embedding persistence diagrams in a Hilbert space.
method Directly use Figalli-Gigli distance to build a positive definite kernel.
result SFGK shares properties with SWK and performs similarly on benchmarks.
Optimizes wavelets for graph classification using spectral wavelet signatures and persistence diagrams.
problem Graph classification with geometric properties encoded in persistence diagrams.
method Optimizes spectral wavelets for graph datasets to capture best-suited features for classification.
result Competitive performance in graph classification problems compared to other persistence-based architectures.
PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signaturesstat.ML
Persistence diagrams, the most common descriptors of Topological Data Analysis, encode topological properties of data and have already proved pivotal in many different applications of data science. However, since the (metric) space of persistence diagrams is not Hilbert, they end up being difficult inputs for most Mach…
Paper introduces DP TDA for near-optimal private persistence diagrams.
problem Challenges in privatizing topological data analysis.
method Sensitivity analysis of persistence diagrams, use of exponential mechanism.
result Proposes near-optimal privacy mechanism for TDA.
Since persistence diagrams do not admit an inner product structure, a map into a Hilbert space is needed in order to use kernel methods. It is natural to ask if such maps necessarily distort the metric on persistence diagrams. We show that persistence diagrams with the bottleneck distance do not even admit a coarse emb…
A method for vectorizing persistence diagrams simplifies topological data analysis.
problem Challenges in integrating persistence diagrams into machine learning pipelines.
method Quantized Persistence and Integral transforms of Diagrams (Qupid) using binning and discrete transforms.
result Qupid preserves highly competitive performances compared to state-of-the-art methods across various classification tasks.
New lattice path method for statistical inference of persistent diagrams.
problem Statistical inference on persistent diagrams.
method Lattice path representation and combinatorial enumerations.
result Topological changes observed in spike proteins of COVID-19 virus.
Maps persistence diagrams into Hilbert and Euclidean spaces with explicit distortions.
problem Embedding persistence diagrams into Euclidean spaces for statistical analysis.
method Explicit geometric maps with distortion functions.
result Controlled geometric information loss through explicit distortion functions.
Topological data analysis is an emerging mathematical concept for characterizing shapes in multi-scale data. In this field, persistence diagrams are widely used as a descriptor of the input data, and can distinguish robust and noisy topological properties. Nowadays, it is highly desired to develop a statistical framewo…
Generates random persistence diagrams for data analysis.
problem Generating random persistence diagrams for data analysis.
method Based on pairwise interacting point processes and RJ-MCMC algorithm.
result Demonstrates the efficacy and utility of RPDG in materials science.
The persistence diagram is an increasingly useful tool from Topological Data Analysis, but its use alongside typical machine learning techniques requires mathematical finesse. The most success to date has come from methods that map persistence diagrams into vector spaces, in a way which maximizes the structure preserve…
This work incorporates topological features via persistence diagrams to classify point cloud data arising from materials science. Persistence diagrams are multisets summarizing the connectedness and holes of given data. A new distance on the space of persistence diagrams generates relevant input features for a classifi…
Persistence diagrams of random matrices via Morse theory: universality and a new spectral diagnosticstat.ML
Persistence diagrams from random matrices follow RMT universality, offering a new spectral diagnostic.
problem Understanding spectral properties of random matrices using topological data analysis.
method Applying Morse theory to persistence diagrams of quadratic forms restricted to unit spheres.
result Persistence entropy outperforms traditional level spacing ratios in discriminating random matrix ensembles.
Persistence diagrams are two-dimensional plots that summarize the topological features of functions and are an important part of topological data analysis. A problem that has received much attention is how deal with sets of persistence diagrams. How do we summarize them, average them or cluster them? One approach -- th…
We prove that the space of persistence diagrams on points (with the bottleneck or a Wasserstein distance) coarsely embeds into Hilbert space by showing it is of asymptotic dimension . Such an embedding enables utilisation of Hilbert space techniques on the space of persistence diagrams. We also prove that when …
STRAND: A single representation for hypothesis testing and vectorisation of persistence diagrams
problem Comparing persistence diagrams
method Survival topological representation analysis
result Non-parametric two-sample test with calibrated Type I error and high power
Study shows how non-uniform scaling affects persistence diagrams.
problem Stability of persistence diagrams under non-uniform scaling.
method Explicit bounds on bottleneck distance derived for Euclidean scaling.
result Explicit bounds on the stability of persistence diagrams under non-uniform scaling.
Unreduced PDs can perform similarly to reduced PDs in machine learning tasks.
problem Ignoring much of the information in persistence diagrams in machine learning pipelines.
method Developed methods to generate topological feature vectors from unreduced boundary matrices.
result Unreduced PDs can perform on par with, and sometimes outperform, fully-reduced PDs in machine learning tasks.
FCM clustering adapts to persistence diagrams for topological data analysis.
problem Integrating topological data into machine learning workflows.
method Adapting Fuzzy c-Means to persistence diagrams.
result FCM clustering captures topological structure without additional processing.
Approaches for approximating persistent homology for large datasets.
problem Inability to compute persistent homology for large datasets.
method Multiple subsampling framework for statistical approximation of persistent homology.
result Derivation of finite sample convergence rates for empirical means of persistent homology.
Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine lea…
Bayesian method classifies actin cytoskeleton networks using topological data.
problem Classifying the structure of biological networks, especially actin cytoskeleton networks.
method Transform actin cytoskeleton networks into persistence diagrams, quantify variability with Bayesian framework, estimate posterior distributions.
result Bayesian framework successfully classifies actin filament networks, outperforming state-of-the-art methods.
We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric descriptions of the "life span" or "persistence" of a homology class. As a case study, …
A new method uses vectorized summaries of persistence diagrams for efficient hypothesis testing.
problem Efficient hypothesis testing for large and complex persistence diagrams.
method Vectorized summaries of Betti functions and a new shuffling technique.
result The vectorized Betti function leads to competitive results compared to baseline methods.
This article addresses persistent tangles. These are tangles whose presence in a knot diagram forces that diagram to be knotted. We provide new methods for constructing persistent tangles. Our techniques rely mainly on the existence of non-trivial colorings for the tangles in question. Our main result in this article i…
Recently many efforts have been made to incorporate persistence diagrams, one of the major tools in topological data analysis (TDA), into machine learning pipelines. To better understand the power and limitation of persistence diagrams, we carry out a range of experiments on both graph data and shape data, aiming to de…
Adaptive template systems: Data-driven feature selection for learning with persistence diagramscs.LG
Feature extraction from persistence diagrams, as a tool to enrich machine learning techniques, has received increasing attention in recent years. In this paper we explore an adaptive methodology to localize features in persistent diagrams, which are then used in learning tasks. Specifically, we investigate three algori…
Estimates and quantizes expected persistence diagrams for efficient analysis.
problem Statistical summary of the topology of structured data.
method Expected Persistence Diagram (EPD) and its quantization.
result Optimal estimation of EPD with near-optimal quantization.
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a …
Improved modeling of persistence diagrams for data analysis.
problem Determining significant outliers in persistence diagrams.
method Modification of the RST (Replicating Statistical Topology) model using MCMC Metropolis-Hastings algorithm.
result The modified RST model improves the goodness of fit in persistence diagram analysis.
A faster, more stable method for optimizing topological functions.
problem Optimizing topological functions is computationally expensive and unstable.
method Introduces a novel backpropagation scheme for faster and more robust optimization.
result Produces more robust optima and stable visualizations.
We define a simple obstruction to Yu's property A that we call -prisms. This structure allows for a straightforward proof that the space of persistence diagrams fails to have property A in a Wasserstein metric.
Many attempts have been made in recent decades to integrate machine learning (ML) and topological data analysis. A prominent problem in applying persistent homology to ML tasks is finding a vector representation of a persistence diagram (PD), which is a summary diagram for representing topological features. From the pe…
Stability and Machine Learning Applications of Persistent Homology Using the Delaunay-Rips Complexstat.CO
Paper defines and evaluates DR complex for persistent homology.
problem Computing persistent homology of Euclidean point cloud data.
method Delaunay-Rips complex construction for speed and stability.
result DR produces stable persistence diagrams under point cloud perturbations.
Persistent Legendrian contact homology distinguishes knots using height functional.
problem Distinguishing Legendrian knots in .
method Persistent homology applied to Chekanov-Eliashberg DGA, with height functional.
result Strong Morse inequalities for persistent Legendrian contact homology.
Unified pipeline classifies time series using complex networks and persistent homology.
problem Classifying univariate time series using various graph constructions and metrics.
method Time series to graph, graph to dissimilarity matrix, filtration to persistence diagrams, vectorization to features.
result Persistence-based features are robust to noise and optimal graph type depends on signal structure.
New approach uses distributed persistence for stable, parallelizable topological analysis of large point clouds.
problem Estimating the full persistence diagram of large point clouds is expensive, unstable, and not a sufficient statistic.
method Proposes distributed persistence as a new invariant, which is perfectly parallelizable, more stable, and has a rich inverse theory.
result The map from point clouds to distributed persistence invariants is a global quasi-isometry, interpolating between purely geometric and topological invariants.
Paper stabilizes persistent homology rank functions for statistical inference.
problem Stability issues in persistent homology rank functions.
method Derive stability results for rank functions under FDA metrics.
result Rank functions stabilize, improving statistical inference.
Study cosmic structures using Topological Data Analysis and Persistence Energy.
problem Investigate cosmic web evolution in CDM cosmologies.
method Apply LITE method to embed persistence diagrams into vector spaces and analyze cosmic structures.
result Discover a correlation between Persistence Energy and redshift values.
Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs exhibit, however, complex structure and are difficult to integrate in today's machine learning workflows. This paper introduces persistence bag-of-words: a novel and stable…
The paper tackles binary classification with measure data using topological descriptors.
problem Binary classification with measure data.
method Develops classifiers for measure data using topological descriptors (persistence diagrams).
result Upper and lower bounds on the Rademacher complexity of classifiers on measures.
This note contributes to the point calculus of persistent homology by extending Alexander duality to real-valued functions. Given a perfect Morse function and a decomposition such that $M = \U \cap V$ is an -manifold, we prove elementary relationships between the persisten…
We introduce the concept of community trees that summarizes topological structures within a network. A community tree is a tree structure representing clique communities from the clique percolation method (CPM). The community tree also generates a persistent diagram. Community trees and persistent diagrams reveal topol…
Shape-Preserving Dimensionality Reduction : An Algorithm and Measures of Topological Equivalencestat.ML
An algorithm preserves topological features in dimensionality reduction.
problem Preserving topological features in dimensionality reduction.
method Simulated annealing for finding a linear projection preserving persistent homology.
result Measures of topological equivalence between filtrations.