Persistence Diagrams Research Articles

Field Choice Problem in Persistent Homology

This paper tackles the problem of coefficient field choice in persistent homology. When we compute a persistence diagram, we need to select a coefficient field before computation. We should understand the dependence of the diagram on the coefficient field to facilitate computation and interpretation of the diagram. We clarify that the dependence is strongly related to the torsion part of mathbb {Z} relative homology in the filtration. We show the sufficient and necessary conditions of the independence of coefficient field choice. An efficient algorithm is proposed to verify the independence. A slight modification of the standard persistence algorithm gives the verification algorithm. In a numerical experiment with the algorithm, a persistence diagram rarely changes even when the coefficient field changes if we consider a filtration in mathbb {R}^3. The experiment suggests that, in practical terms, changes in the field coefficient will not change persistence diagrams when the data are in mathbb {R}^3.

Statistical Embedding: Beyond Principal Components

There has been an intense recent activity in embedding of very high-dimensional and nonlinear data structures, much of it in the data science and machine learning literature. We survey this activity in four parts. In the first part, we cover nonlinear methods such as principal curves, multidimensional scaling, local linear methods, ISOMAP, graph-based methods and diffusion mapping, kernel based methods and random projections. The second part is concerned with topological embedding methods, in particular mapping topological properties into persistence diagrams and the Mapper algorithm. Another type of data sets with a tremendous growth is very high-dimensional network data. The task considered in part three is how to embed such data in a vector space of moderate dimension to make the data amenable to traditional techniques such as cluster and classification techniques. Arguably, this is the part where the contrast between algorithmic machine learning methods and statistical modeling, represented by the so-called stochastic block model, is at its greatest. In the paper, we discuss the pros and cons for the two approaches. The final part of the survey deals with embedding in R2, that is, visualization. Three methods are presented: t-SNE, UMAP and LargeVis based on methods in parts one, two and three, respectively. The methods are illustrated and compared on two simulated data sets; one consisting of a triplet of noisy Ranunculoid curves, and one consisting of networks of increasing complexity generated with stochastic block models and with two types of nodes.

A Probabilistic Result on Impulsive Noise Reduction in Topological Data Analysis through Group Equivariant Non-Expansive Operators.

In recent years, group equivariant non-expansive operators (GENEOs) have started to find applications in the fields of Topological Data Analysis and Machine Learning. In this paper we show how these operators can be of use also for the removal of impulsive noise and to increase the stability of TDA in the presence of noisy data. In particular, we prove that GENEOs can control the expected value of the perturbation of persistence diagrams caused by uniformly distributed impulsive noise, when data are represented by L-Lipschitz functions from R to R.

A universal null-distribution for topological data analysis

One of the most elusive challenges within the area of topological data analysis is understanding the distribution of persistence diagrams arising from data. Despite much effort and its many successful applications, this is largely an open problem. We present a surprising discovery: normalized properly, persistence diagrams arising from random point-clouds obey a universal probability law. Our statements are based on extensive experimentation on both simulated and real data, covering point-clouds with vastly different geometry, topology, and probability distributions. Our results also include an explicit well-known distribution as a candidate for the universal law. We demonstrate the power of these new discoveries by proposing a new hypothesis testing framework for computing significance values for individual topological features within persistence diagrams, providing a new quantitative way to assess the significance of structure in data.

The robustness of persistent homology of brain networks to data acquisition-related non-neural variability in resting state fMRI.

There is increasing interest in investigating brain function based on functional connectivity networks (FCN) obtained from resting-state functional magnetic resonance imaging (fMRI). FCNs, typically obtained using measures of time series association such as Pearson's correlation, are sensitive to data acquisition parameters such as sampling period. This introduces non-neural variability in data pooled from different acquisition protocols and MRI scanners, negating the advantages of larger sample sizes in pooled data. To address this, we hypothesize that the topology or shape of brain networks must be preserved irrespective of how densely it is sampled, and metrics which capture this topology may be statistically similar across sampling periods, thereby alleviating this source of non-neural variability. Accordingly, we present an end-to-end pipeline that uses persistent homology (PH), a branch of topological data analysis, to demonstrate similarity across FCNs acquired at different temporal sampling periods. PH, as a technique, extracts topological features by capturing the network organization across all continuous threshold values, as opposed to graph theoretic methods, which fix a discrete network topology by thresholding the connectivity matrix. The extracted topological features are encoded in the form of persistent diagrams that can be compared against one another using the earth-moving metric, also popularly known as the Wasserstein distance. We extract topological features from three data cohorts, each acquired at different temporal sampling periods and demonstrate that these features are statistically the same, hence, empirically showing that PH may be robust to changes in data acquisition parameters such as sampling period.

Reasonable thickness determination for implicit porous sheet structure using persistent homology

Porous structures are widely used in various industries because of their excellent properties. Porous surfaces have no thickness and should be thickened to sheet structures for further fabrication. For porous surfaces with implicit representation, sheet structures can be generated by extracting the region between two iso-surfaces. However, too small sheet thickness leads to extra small holes emerging on the sheet structure, and too large sheet thickness makes the pores closed. In this study, we propose a novel method for determining the sheet thickness for porous surfaces represented by implicit functions. Specifically, after calculating the persistent diagram (PD) of the scalar field dictated by the implicit function, the points in the PD are clustering into three clusters. Using these clusters, a topology control method is developed, which allows users to determine how many pores can be closed. Moreover, a reasonable thickness can be determined, by which all of the pores keep open, and the extra small holes are eliminated. Experimental results show the effectiveness of the developed method.

Stability and machine learning applications of persistent homology using the Delaunay-Rips complex

Persistent homology (PH) is a robust method to compute multi-dimensional geometric and topological features of a dataset. Because these features are often stable under certain perturbations of the underlying data, are often discriminating, and can be used for visualization of structure in high-dimensional data and in statistical and machine learning modeling, PH has attracted the interest of researchers across scientific disciplines and in many industry applications. However, computational costs may present challenges to effectively using PH in certain data contexts, and theoretical stability results may not hold in practice. In this paper, we define, implement, and investigate a simplicial complex construction for computing persistent homology of Euclidean point cloud data, which we call the Delaunay-Rips complex (DR). By only considering simplices that appear in the Delaunay triangulation of the point cloud and assigning the Vietoris-Rips weights to simplices, DR avoids potentially costly computations in the persistence calculations. We document and compare a Python implementation of DR with other simplicial complex constructions for generating persistence diagrams. By imposing sufficient conditions on point cloud data, we are able to theoretically justify the stability of the persistence diagrams produced using DR. When the Delaunay triangulation of the point cloud changes under perturbations of the points, we prove that DR-produced persistence diagrams exhibit instability. Since we cannot guarantee that real-world data will satisfy our stability conditions, we demonstrate the practical robustness of DR for persistent homology in comparison with other simplicial complexes in machine learning applications. We find in our experiments that using DR in an ML-TDA pipeline performs comparatively well as using other simplicial complex constructions.

Persistent Homology through Image Segmentation (Student Abstract)

The efficacy of topological data analysis (TDA) has been demonstrated in many different machine learning pipelines, particularly those in which structural characteristics of data are highly relevant. However, TDA's usability in large scale machine learning applications is hindered by the significant computational cost of generating persistence diagrams. In this work, a method that allows this computationally expensive process to be approximated by deep neural networks is proposed. Moreover, the method's practicality in estimating 0-dimensional persistence diagrams across a diverse range of images is shown.

An integrated image visibility graph and topological data analysis for extracting time series features

A time series can often be characterized using machine learning techniques, which require feature vectors as input. The quality of the feature vectors reflects the accuracy of the utilized machine learning techniques. We propose a method for combining features extracted from two popular techniques, one from the persistence diagram, a summary of topological data analysis, and another from the visibility graph, a popular approach for constructing complex networks. We further present a method for extracting features from time series by transforming the series into a two-dimensional array and applying a 2-dimensional visibility graph. The 2-dimensional visibility graph improves the quality of the features for clustering. The features extracted from the techniques are utilized for clustering 1100 time series samples simulated from 11 classes. The two clustering quality evaluation metrics namely normalized mutual info score and adjusted rand score, which are scores given to clustering algorithms, are used to measure the clustering quality. Inclusion of 2-dimensional visibility graph features increases the values of these metrics for classifying the time series. Therefore, the results confirm the usefulness of including a 2-dimensional visibility graph from time series clustering and analysis.

Geometric characterization of the persistence of 1D maps

We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining a collection of sorted lists together with its persistence diagram.

Topological Approach for Material Structure Analyses in Terms of R2 Orientation Distribution Function

The application of solid mechanics theory for material behavior faces the discrete nature of modern or biological material. Despite the developed methods of homogenization, there are deviations between simulated and experiments results. The reason is homogenization, which mathematically involves a type of interpolation. The situation gets worse for complex structured materials. On the other hand, a topological approach can help in such analysis, but such an approach has computational costs. At the same time, increasing modern computational capabilities remove this barrier. This study is focused on building a method to analyze material structure in a topological sense. The orientation distribution function was used to describe the structure of the material. The plane case was investigated. Quadratic and biquadratic forms of interpolant were investigated. The persistent homology approach was used for topology analysis. For this purpose, a persistence diagram for quadratic and biquadratic forms was found and analyzed. In this study, it is shown how scaling the origin point cloud influences H1 points in the persistence diagram. It was assumed that the topology of the biquadratic form can be understood as a superposition of quadratic forms. Quantitative estimates are given for ellipticity and H1 points. A dataset of micro photos was processed using the proposed method. Furthermore, the supply criteria for the interpolation choice in quadratic or biquadratic forms was formulated.

Formation of features based on computational topology methods

The use of traditional methods of algebraic topology to obtain information about the shape of an object is associated with the problem of forming a small amount of information, namely, Betti numbers and Euler characteristics. The central tool for topological data analysis is the persistent homology method, which summarizes the geometric and topological information in the data using persistent diagrams and barcodes. Based on persistent homology methods, topological data can be analyzed to obtain information about the shape of an object. The construction of persistent barcodes and persistent diagrams in computational topology does not allow one to construct a Hilbert space with a scalar product. The possibility of applying the methods of topological data analysis is based on mapping persistent diagrams into a Hilbert space; one of the ways of such mapping is a method of constructing a persistence landscape. It has an advantage of being reversible, so it does not lose any information and has persistence properties. The paper considers mathematical models and functions for representing persistence landscape objects based on the persistent homology method. Methods for converting persistent barcodes and persistent diagrams into persistence landscape functions are considered. Associated with persistence landscape functions is a persistence landscape kernel that forms a mapping into a Hilbert space with a dot product. A formula is proposed for determining a distance between the persistence landscapes, which allows the distance between images of objects to be found. The persistence landscape functions map persistent diagrams into a Hilbert space. Examples of determining the distance between images based on the construction of persistence landscape functions for these images are given. Representations of topological characteristics in various models of computational topology are considered. Results for one-parameter persistence modules are extended onto multi-parameter persistence modules.

Feature Construction Using Persistence Landscapes for Clustering Noisy IoT Time Series

With the advancement of IoT technologies, there is a large amount of data available from wireless sensor networks (WSN), particularly for studying climate change. Clustering long and noisy time series has become an important research area for analyzing this data. This paper proposes a feature-based clustering approach using topological data analysis, which is a set of methods for finding topological structure in data. Persistence diagrams and landscapes are popular topological summaries that can be used to cluster time series. This paper presents a framework for selecting an optimal number of persistence landscapes, and using them as features in an unsupervised learning algorithm. This approach reduces computational cost while maintaining accuracy. The clustering approach was demonstrated to be accurate on simulated data, based on only four, three, and three features, respectively, selected in Scenarios 1–3. On real data, consisting of multiple long temperature streams from various US locations, our optimal feature selection method achieved approximately a 13 times speed-up in computing.

Adaptive Slicing of Implicit Porous Structure with Topology Guarantee

Adaptive slicing is an important step in 3D printing, where the thicknesses of slices are adaptively adjusted to achieve a trade-off between printing time reduction and surface quality improvement. However, in conventional slicing algorithms, the topological information inside the model is usually not considered, and then, they are not suitable for slicing porous structures with complicated inner topological structures. In this study, based on the persistent homology theory, a novel method is developed to adaptively slice implicit porous structures, which guarantees the topological correctness of the generated adaptive slice model. Given an implicit porous structure, we generate the finest slice model, and calculate the topological information of the finest slices, i.e., persistent diagram (PD) and Betti number. Next, the finest slice model is partitioned into parts based on the Betti number of the finest slices. In each part, the Betti numbers of the finest slices are the same. Moreover, in each part, we first determine the slices for printing by solving the ℓ0−norm optimization problem to optimize the printing time, and then, the thicknesses of the slices for printing are calculated by optimizing the surface quality. In this way, not only can the printing time and surface quality be optimized, but also the topological correctness of the adaptive slice model can be guaranteed. Experimental results show the effectiveness of the developed algorithm.

Feature extraction based on time-series topological analysis for the partial discharge pattern recognition of high-voltage power cables

In the partial discharge (PD) pattern recognition of power cables, the existing time–frequency features often exert an impact on recognition accuracy because of insufficient discrimination. A novel PD feature extraction and identification method on the basis of time-series topological data analysis (TDA) was proposed in this paper. Firstly, original PD sequence was reconstructed as point cloud in phase space based on optimized symbolic entropy. Then, a PD topological space is constituted with point cloud to extract its persistent homology features. On this basis, persistence diagrams and barcodes were calculated and visually expressed as Betty curves. Finally, Betty curves were input into an optimized 1D convolution neural network (1D-CNN) model to recognize four typical PD patterns and carry out comparison experiments. The visualization produced by t-distributed stochastic neighbor embedding (t-SNE) shows that TDA features possess significant discrimination, experiencing an increase of 11.25% in the overall recognition accuracy and reaching 98.00% compared with original PD sequence and time–frequency features. Meanwhile, the computation cost of the proposed algorithm is optimized within the permissible range for real-time applications.

Bottleneck Profiles and Discrete Prokhorov Metrics for Persistence Diagrams

In topological data analysis (TDA), persistence diagrams (PDs) have been a successful tool. To compare them, Wasserstein and bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in a systematic way by introducing bottleneck profiles. This leads to a notion of discrete Prokhorov metrics for PDs as a generalization of the bottleneck distance. These metrics satisfy a stability result and can be used to bound Wasserstein metrics from above and from below. We provide algorithms to compute the newly introduced quantities and end with an discussion about experiments.

Stable volumes for persistent homology

This paper proposes a stable volume and a stable volume variant, referred to as a stable sub-volume, for more reliable data analysis using persistent homology. In prior research, an optimal cycle and similar ideas have been proposed to identify the homological structure corresponding to each birth-death pair in a persistence diagram. While this is helpful for data analysis using persistent homology, the results are sensitive to noise. The sensitivity affects the reliability and interpretability of the analysis. In this paper, stable volumes and stable sub-volumes are proposed to solve this problem. For a special case, we prove that a stable volume is the robust part of an optimal volume against noise. We implemented stable volumes and sub-volumes on HomCloud, a data analysis software package based on persistent homology, and show examples of stable volumes and sub-volumes.

Cycle Registration in Persistent Homology With Applications in Topological Bootstrap.

We propose a novel approach for comparing the persistent homology representations of two spaces (or filtrations). Commonly used methods are based on numerical summaries such as persistence diagrams and persistence landscapes, along with suitable metrics (e.g., Wasserstein). These summaries are useful for computational purposes, but they are merely a marginal of the actual topological information that persistent homology can provide. Instead, our approach compares between two topological representations directly in the data space. We do so by defining a correspondence relation between individual persistent cycles of two different spaces, and devising a method for computing this correspondence. Our matching of cycles is based on both the persistence intervals and the spatial placement of each feature. We demonstrate our new framework in the context of topological inference, where we use statistical bootstrap methods in order to differentiate between real features and noise in point cloud data.

A topological loss function for image Denoising on a new BVI-lowlight dataset

Although image denoising algorithms have attracted significant research attention, surprisingly few have been proposed for, or evaluated on, noise from imagery acquired under real low-light conditions. Moreover, noise characteristics are often assumed to be spatially invariant, leading to edges and textures being distorted after denoising. Here, we introduce a novel topological loss function which is based on persistent homology. The method performs in the space of image patches, where topological invariants are calculated and represented in persistent diagrams. The loss function is a combination of ℓ1 or ℓ2 losses with the new persistence-based topological loss. We compare its performance across popular denoising architectures and loss functions, training the networks on our new comprehensive dataset of natural images captured in low-light conditions – BVI-LOWLIGHT. Analysis reveals that this approach outperforms existing methods, adapting well to complex structures and suppressing common artifacts.

Feature Detection and Hypothesis Testing for Extremely Noisy Nanoparticle Images using Topological Data Analysis

ABSTRACT We propose a flexible algorithm for feature detection and hypothesis testing in images with ultra-low signal-to-noise ratio using cubical persistent homology. Our main application is in the identification of atomic columns and other features in Transmission Electron Microscopy (TEM). Cubical persistent homology is used to identify local minima and their size in subregions in the frames of nanoparticle videos, which are hypothesized to correspond to relevant atomic features. We compare the performance of our algorithm to other employed methods for the detection of columns and their intensity. Additionally, Monte Carlo goodness-of-fit testing using real-valued summaries of persistence diagrams derived from smoothed images (generated from pixels residing in the vacuum region of an image) is developed and employed to identify whether or not the proposed atomic features generated by our algorithm are due to noise. Using these summaries derived from the generated persistence diagrams, one can produce univariate time series for the nanoparticle videos, thus, providing a means for assessing fluxional behavior. A guarantee on the false discovery rate for multiple Monte Carlo testing of identical hypotheses is also established.