Topological features in signal processing using frame theory and persistent homology

2017 ◽  
Vol 141 (5) ◽  
pp. 3585-3585
Author(s):  
Mijail Guillemard
Keyword(s):  
Signal Processing ◽  
Persistent Homology ◽  
Frame Theory ◽  
Topological Features

scholarly journals Effect of APOE ε4 on multimodal brain connectomic traits: a persistent homology study

2020 ◽  
Vol 21 (S21) ◽  
Author(s):  
Jin Li ◽  
◽  
Chenyuan Bian ◽  
Dandan Chen ◽  
Xianglian Meng ◽  
...  
Keyword(s):  
Magnetic Resonance Imaging ◽  
Magnetic Resonance ◽  
Genetic Risk ◽  
Brain Connectivity ◽  
Persistent Homology ◽  
Brain Network ◽  
Resonance Imaging ◽  
Modeling Framework ◽  
Apoe Ε4 ◽  
Topological Features

Abstract Background Although genetic risk factors and network-level neuroimaging abnormalities have shown effects on cognitive performance and brain atrophy in Alzheimer’s disease (AD), little is understood about how apolipoprotein E (APOE) ε4 allele, the best-known genetic risk for AD, affect brain connectivity before the onset of symptomatic AD. This study aims to investigate APOE ε4 effects on brain connectivity from the perspective of multimodal connectome. Results Here, we propose a novel multimodal brain network modeling framework and a network quantification method based on persistent homology for identifying APOE ε4-related network differences. Specifically, we employ sparse representation to integrate multimodal brain network information derived from both the resting state functional magnetic resonance imaging (rs-fMRI) data and the diffusion-weighted magnetic resonance imaging (dw-MRI) data. Moreover, persistent homology is proposed to avoid the ad hoc selection of a specific regularization parameter and to capture valuable brain connectivity patterns from the topological perspective. The experimental results demonstrate that our method outperforms the competing methods, and reasonably yields connectomic patterns specific to APOE ε4 carriers and non-carriers. Conclusions We have proposed a multimodal framework that integrates structural and functional connectivity information for constructing a fused brain network with greater discriminative power. Using persistent homology to extract topological features from the fused brain network, our method can effectively identify APOE ε4-related brain connectomic biomarkers.

scholarly journals A topological characterization of DNA sequences based on chaos geometry and persistent homology

2021 ◽  
Author(s):  
Dong Quan Ngoc Nguyen ◽  
Phuong Dong Tan Le ◽  
Lin Xing ◽  
Lizhen Lin
Keyword(s):  
Dna Sequences ◽  
Algebraic Topology ◽  
Influenza A ◽  
Graphical Representation ◽  
Dimensional Space ◽  
Persistent Homology ◽  
Point Clouds ◽  
Influenza A Viruses ◽  
Topological Characterization ◽  
Topological Features

AbstractMethods for analyzing similarities among DNA sequences play a fundamental role in computational biology, and have a variety of applications in public health, and in the field of genetics. In this paper, a novel geometric and topological method for analyzing similarities among DNA sequences is developed, based on persistent homology from algebraic topology, in combination with chaos geometry in 4-dimensional space as a graphical representation of DNA sequences. Our topological framework for DNA similarity analysis is general, alignment-free, and can deal with DNA sequences of various lengths, while proving first-of-the-kind visualization features for visual inspection of DNA sequences directly, based on topological features of point clouds that represent DNA sequences. As an application, we test our methods on three datasets including genome sequences of different types of Hantavirus, Influenza A viruses, and Human Papillomavirus.

scholarly journals Bot Detection on Social Networks Using Persistent Homology

2020 ◽  
Vol 25 (3) ◽  
pp. 58
Author(s):  
Minh Nguyen ◽  
Mehmet Aktas ◽  
Esra Akbas
Keyword(s):  
Machine Learning ◽  
Social Networks ◽  
Social Media ◽  
Persistent Homology ◽  
Structural Features ◽  
Higher Order ◽  
Ego Networks ◽  
Feature Extraction Method ◽  
Topological Features ◽  
Bot Detection

The growth of social media in recent years has contributed to an ever-increasing network of user data in every aspect of life. This volume of generated data is becoming a vital asset for the growth of companies and organizations as a powerful tool to gain insights and make crucial decisions. However, data is not always reliable, since primarily, it can be manipulated and disseminated from unreliable sources. In the field of social network analysis, this problem can be tackled by implementing machine learning models that can learn to classify between humans and bots, which are mostly harmful computer programs exploited to shape public opinions and circulate false information on social media. In this paper, we propose a novel topological feature extraction method for bot detection on social networks. We first create weighted ego networks of each user. We then encode the higher-order topological features of ego networks using persistent homology. Finally, we use these extracted features to train a machine learning model and use that model to classify users as bot vs. human. Our experimental results suggest that using the higher-order topological features coming from persistent homology is promising in bot detection and more effective than using classical graph-theoretic structural features.

scholarly journals Using Topological Data Analysis (TDA) and Persistent Homology to Analyze the Stock Markets in Singapore and Taiwan

2021 ◽  
Vol 9 ◽  
Author(s):  
Peter Tsung-Wen Yen ◽  
Siew Ann Cheong
Keyword(s):  
Data Analysis ◽  
Stock Markets ◽  
Persistent Homology ◽  
Betti Numbers ◽  
Stock Index ◽  
Topological Data Analysis ◽  
Series Data ◽  
Topological Features ◽  
Market Crashes ◽  
Topological Data

In recent years, persistent homology (PH) and topological data analysis (TDA) have gained increasing attention in the fields of shape recognition, image analysis, data analysis, machine learning, computer vision, computational biology, brain functional networks, financial networks, haze detection, etc. In this article, we will focus on stock markets and demonstrate how TDA can be useful in this regard. We first explain signatures that can be detected using TDA, for three toy models of topological changes. We then showed how to go beyond network concepts like nodes (0-simplex) and links (1-simplex), and the standard minimal spanning tree or planar maximally filtered graph picture of the cross correlations in stock markets, to work with faces (2-simplex) or any k-dim simplex in TDA. By scanning through a full range of correlation thresholds in a procedure called filtration, we were able to examine robust topological features (i.e. less susceptible to random noise) in higher dimensions. To demonstrate the advantages of TDA, we collected time-series data from the Straits Times Index and Taiwan Capitalization Weighted Stock Index (TAIEX), and then computed barcodes, persistence diagrams, persistent entropy, the bottleneck distance, Betti numbers, and Euler characteristic. We found that during the periods of market crashes, the homology groups become less persistent as we vary the characteristic correlation. For both markets, we found consistent signatures associated with market crashes in the Betti numbers, Euler characteristics, and persistent entropy, in agreement with our theoretical expectations.

scholarly journals Cubical homology-based Image Classification - A Comparative Study

2021 ◽  
Author(s):  
◽  
Seungho Choe
Keyword(s):  
Machine Learning ◽  
Image Classification ◽  
Digital Image ◽  
Persistent Homology ◽  
Topological Data Analysis ◽  
Connected Components ◽  
Gradient Boosting ◽  
Topological Features ◽  
Light Gradient ◽  
Cubical Homology

Persistent homology is a powerful tool in topological data analysis (TDA) to compute, study and encode efficiently multi-scale topological features and is being increasingly used in digital image classification. The topological features represent number of connected components, cycles, and voids that describe the shape of data. Persistent homology extracts the birth and death of these topological features through a filtration process. The lifespan of these features can represented using persistent diagrams (topological signatures). Cubical homology is a more efficient method for extracting topological features from a 2D image and uses a collection of cubes to compute the homology, which fits the digital image structure of grids. In this research, we propose a cubical homology-based algorithm for extracting topological features from 2D images to generate their topological signatures. Additionally, we propose a score, which measures the significance of each of the sub-simplices in terms of persistence. Also, gray level co-occurrence matrix (GLCM) and contrast limited adapting histogram equalization (CLAHE) are used as a supplementary method for extracting features. Machine learning techniques are then employed to classify images using the topological signatures. Among the eight tested algorithms with six published image datasets with varying pixel sizes, classes, and distributions, our experiments demonstrate that cubical homology-based machine learning with deep residual network (ResNet 1D) and Light Gradient Boosting Machine (lightGBM) shows promise with the extracted topological features.

scholarly journals Gene Coexpression Network Comparison via Persistent Homology

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Ali Nabi Duman ◽  
Harun Pirim
Keyword(s):  
Microarray Data ◽  
Clustering Algorithm ◽  
Persistent Homology ◽  
Distance Matrix ◽  
Stress Factors ◽  
Topological Data Analysis ◽  
Data Sets ◽  
Computationally Efficient ◽  
Topological Features ◽  
Gene Coexpression

Persistent homology, a topological data analysis (TDA) method, is applied to microarray data sets. Although there are a few papers referring to TDA methods in microarray analysis, the usage of persistent homology in the comparison of several weighted gene coexpression networks (WGCN) was not employed before to the very best of our knowledge. We calculate the persistent homology of weighted networks constructed from 38 Arabidopsis microarray data sets to test the relevance and the success of this approach in distinguishing the stress factors. We quantify multiscale topological features of each network using persistent homology and apply a hierarchical clustering algorithm to the distance matrix whose entries are pairwise bottleneck distance between the networks. The immunoresponses to different stress factors are distinguishable by our method. The networks of similar immunoresponses are found to be close with respect to bottleneck distance indicating the similar topological features of WGCNs. This computationally efficient technique analyzing networks provides a quick test for advanced studies.

scholarly journals Minimal Cycle Representatives in Persistent Homology Using Linear Programming: An Empirical Study With User’s Guide

2021 ◽  
Vol 4 ◽  
Author(s):  
Lu Li ◽  
Connor Thompson ◽  
Gregory Henselman-Petrusek ◽  
Chad Giusti ◽  
Lori Ziegelmeier
Keyword(s):  
Linear Programming ◽  
Persistent Homology ◽  
Computational Cost ◽  
Computation Time ◽  
Integer Program ◽  
General Purpose ◽  
Cloud Data ◽  
Topological Features ◽  
Minimal Cycle ◽  
Cycle Bases

Cycle representatives of persistent homology classes can be used to provide descriptions of topological features in data. However, the non-uniqueness of these representatives creates ambiguity and can lead to many different interpretations of the same set of classes. One approach to solving this problem is to optimize the choice of representative against some measure that is meaningful in the context of the data. In this work, we provide a study of the effectiveness and computational cost of several ℓ1 minimization optimization procedures for constructing homological cycle bases for persistent homology with rational coefficients in dimension one, including uniform-weighted and length-weighted edge-loss algorithms as well as uniform-weighted and area-weighted triangle-loss algorithms. We conduct these optimizations via standard linear programming methods, applying general-purpose solvers to optimize over column bases of simplicial boundary matrices. Our key findings are: 1) optimization is effective in reducing the size of cycle representatives, though the extent of the reduction varies according to the dimension and distribution of the underlying data, 2) the computational cost of optimizing a basis of cycle representatives exceeds the cost of computing such a basis, in most data sets we consider, 3) the choice of linear solvers matters a lot to the computation time of optimizing cycles, 4) the computation time of solving an integer program is not significantly longer than the computation time of solving a linear program for most of the cycle representatives, using the Gurobi linear solver, 5) strikingly, whether requiring integer solutions or not, we almost always obtain a solution with the same cost and almost all solutions found have entries in {‐1,0,1} and therefore, are also solutions to a restricted ℓ0 optimization problem, and 6) we obtain qualitatively different results for generators in Erdős-Rényi random clique complexes than in real-world and synthetic point cloud data.

scholarly journals Topological exploration of artificial neuronal network dynamics

2018 ◽  
Author(s):  
Jean-Baptiste Bardin ◽  
Gard Spreemann ◽  
Kathryn Hess
Keyword(s):  
Spike Train ◽  
Algebraic Topology ◽  
Persistent Homology ◽  
Network Dynamics ◽  
Learning Classifier ◽  
Topological Features ◽  
Global Properties ◽  
Novel Approach ◽  
Neuronal Network Dynamics ◽  
Artificial Neuronal Network

AbstractOne of the paramount challenges in neuroscience is to understand the dynamics of individual neurons and how they give rise to network dynamics when interconnected. Historically, researchers have resorted to graph theory, statistics, and statistical mechanics to describe the spatiotemporal structure of such network dynamics. Our novel approach employs tools from algebraic topology to characterize the global properties of network structure and dynamics.We propose a method based on persistent homology to automatically classify network dynamics using topological features of spaces built from various spike-train distances. We investigate the efficacy of our method by simulating activity in three small artificial neural networks with different sets of parameters, giving rise to dynamics that can be classified into four regimes. We then compute three measures of spike train similarity and use persistent homology to extract topological features that are fundamentally different from those used in traditional methods. Our results show that a machine learning classifier trained on these features can accurately predict the regime of the network it was trained on and also generalize to other networks that were not presented during training. Moreover, we demonstrate that using features extracted from multiple spike-train distances systematically improves the performance of our method.

scholarly journals Persistent Homology of Tumor CT Scans Predicts Survival In Lung Cancer

2020 ◽  
Author(s):  
Eashwar Somasundaram ◽  
Adam Litzler ◽  
Raoul R. Wadhwa ◽  
Jacob G. Scott
Keyword(s):  
Lung Cancer ◽  
Clinical Oncology ◽  
Lung Tumor ◽  
Persistent Homology ◽  
Cox Proportional Hazards Model ◽  
Ct Scans ◽  
Topological Feature ◽  
Image Size ◽  
Topological Features ◽  
First Moment

ABSTRACTRadiomics, the objective study of non-visual features in clinical imaging, has been useful in informing decisions in clinical oncology. However, radiomics currently lacks the ability to characterize the overall structure of the data. This field may benefit by incorporating persistent homology, a popular new algorithm that analyzes whole data structure. We hypothesized that persistent homology could be applied to lung tumor scans and predict clinical variables. We obtained computed tomography lung scans (n = 565) from the NSCLC-Radiomics and NSCLC-Radiogenomics datasets in The Cancer Imaging Archive. Segmentation data was used to create a cubical region centered on the primary tumor in each scan. For each scan, a cubical complex filtration based on Hounsfield units was generated. We created a feature curve that plotted the number of 0 dimensional topological features against each Hounsfield unit. The curve’s first moment of the distribution was utilized as a summary statistic to predict survival in a Cox proportional hazards model. The first moment of the distribution is equivalent to the area under the curve of our topological feature curves (AUC). The Kruskal-Wallis H Test and a post-hoc Dunn’s test with Bonferroni correction were used to test AUC differences among survival quartiles. After controlling for tumor image size, age, and stage, AUC, was associated with poorer survival (HR = 1.118; 95% CI = 1.026-1.218; p = 0.01). AUC was significantly higher for patients in the lowest survival quartile compared to the highest survival quartile (p < 0.001). We have shown that persistent homology can generate useful clinical correlates from tumor CT scans. Our 0-dimensional topological feature curve statistic predicts survival in lung cancer patients. This novel statistic may be used in tandem with standard radiomics variables to better inform clinical oncology decisions.

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