Persistent homology
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Hierarchical structures of amorphous solids characterized by persistent homology Open
Significance Persistent homology is an emerging mathematical concept for characterizing shapes of data. In particular, it provides a tool called the persistence diagram that extracts multiscale topological features such as rings and caviti…
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TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions Open
weilab.math.msu.edu/TDL/.
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Persistent homology analysis of brain artery trees Open
New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from persisten…
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A Topological Loss Function for Deep-Learning Based Image Segmentation Using Persistent Homology Open
We introduce a method for training neural networks to perform image or volume segmentation in which prior knowledge about the topology of the segmented object can be explicitly provided and then incorporated into the training process. By u…
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Ripser.py: A Lean Persistent Homology Library for Python Open
by computing topological descriptors that summarize features as connected components, loops, and voids.TDA has found wide applications across nonlinear time series analysis (Perea & Harer, 2015), com
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Deep Residual Learning for Compressed Sensing CT Reconstruction via Persistent Homology Analysis Open
Recently, compressed sensing (CS) computed tomography (CT) using sparse projection views has been extensively investigated to reduce the potential risk of radiation to patient. However, due to the insufficient number of projection views, a…
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Persistent homology of time-dependent functional networks constructed from coupled time series Open
We use topological data analysis to study “functional networks” that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into the…
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Localization in the Crowd with Topological Constraints Open
We address the problem of crowd localization, i.e., the prediction of dots corresponding to people in a crowded scene. Due to various challenges, a localization method is prone to spatial semantic errors, i.e., predicting multiple dots wit…
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Persistent Homology Analysis for Materials Research and Persistent Homology Software: HomCloud Open
This paper introduces persistent homology, which is a powerful tool to\ncharacterize the shape of data using the mathematical concept of topology. We\nexplain the fundamental idea of persistent homology from scratch using some\nexamples. W…
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Persistent spectral graph Open
Persistent homology is constrained to purely topological persistence, while multiscale graphs account only for geometric information. This work introduces persistent spectral theory to create a unified low‐dimensional multiscale paradigm f…
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Analysis and prediction of protein folding energy changes upon mutation by element specific persistent homology Open
Supplementary data are available at Bioinformatics online.
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Topological analysis of data Open
Propelled by a fast evolving landscape of techniques and datasets, data science is growing rapidly. Against this background, topological data analysis (TDA) has carved itself a niche for the analysis of datasets that present complex intera…
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Deep Learning with Topological Signatures Open
Inferring topological and geometrical information from data can offer an alternative perspective on machine learning problems. Methods from topological data analysis, e.g., persistent homology, enable us to obtain such information, typical…
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Geometric and Topological Inference Open
Geometric and topological inference deals with the retrieval of information about a geometric object using only a finite set of possibly noisy sample points. It has connections to manifold learning and provides the mathematical and algorit…
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Topological representations of crystalline compounds for the machine-learning prediction of materials properties Open
Accurate theoretical predictions of desired properties of materials play an important role in materials research and development. Machine learning (ML) can accelerate the materials design by building a model from input data. For complex da…
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Persistent homology of complex networks for dynamic state detection Open
In this paper we develop an alternative topological data analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a…
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Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors Open
Significance Quantifying and comparing complex spatial biological datasets is crucial for medical applications and remains an active area of research. As datasets become more heterogeneous and complicated, so must the methods that are used…
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Metrics for comparing neuronal tree shapes based on persistent homology Open
As more and more neuroanatomical data are made available through efforts such as NeuroMorpho.Org and FlyCircuit.org, the need to develop computational tools to facilitate automatic knowledge discovery from such large datasets becomes more …
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PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures Open
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 pers…
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Topological Data Analysis as a Morphometric Method: Using Persistent Homology to Demarcate a Leaf Morphospace Open
Current morphometric methods that comprehensively measure shape cannot compare the disparate leaf shapes found in seed plants and are sensitive to processing artifacts. We explore the use of persistent homology, a topological method applie…
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Persistent homology analysis of craze formation Open
We apply a persistent homology analysis to investigate the behavior of nanovoids during the crazing process of glassy polymers. We carry out a coarse-grained molecular dynamics simulation of the uniaxial deformation of an amorphous polymer…
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Algebraic stability of zigzag persistence modules Open
The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of [math] –valued functions, the result was later cast in a more…
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Topological Persistence for Relating Microstructure and Capillary Fluid Trapping in Sandstones Open
Results from a series of two‐phase fluid flow experiments in Leopard, Berea, and Bentheimer sandstones are presented. Fluid configurations are characterized using laboratory‐based and synchrotron based 3‐D X‐ray computed tomography. All fl…
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Insights into Brain Architectures from the Homological Scaffolds of Functional Connectivity Networks Open
In recent years, the application of network analysis to neuroimaging data has provided useful insights about the brain's functional and structural organization in both health and disease. This has proven a significant paradigm shift from t…
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Volume-Optimal Cycle: Tightest Representative Cycle of a Generator in Persistent Homology Open
The present paper shows a mathematical formalization of---as well as algorithms and software for computing---volume-optimal cycles. Volume-optimal cycles are useful for understanding geometric features appearing in a persistence diagram. V…
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Persistence Paths and Signature Features in Topological Data Analysis Open
We introduce a new feature map for barcodes as they arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values …
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Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization Open
In topological data analysis and visualization, topological descriptors such as persistence diagrams, merge trees, contour trees, Reeb graphs, and Morse–Smale complexes play an essential role in capturing the shape of scalar field data. We…
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The Accumulated Persistence Function, a New Useful Functional Summary Statistic for Topological Data Analysis, With a View to Brain Artery Trees and Spatial Point Process Applications Open
We start with a simple introduction to topological data analysis where the most popular tool is called a persistence diagram. Briefly, a persistence diagram is a multiset of points in the plane describing the persistence of topological fea…
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The Persistent Homology Mathematical Framework Provides Enhanced Genotype-to-Phenotype Associations for Plant Morphology Open
Efforts to understand the genetic and environmental conditioning of plant morphology are hindered by the lack of flexible and effective tools for quantifying morphology. Here, we demonstrate that persistent-homology-based topological metho…
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Topological data analysis quantifies biological nano-structure from single molecule localization microscopy Open
Motivation Localization microscopy data is represented by a set of spatial coordinates, each corresponding to a single detection, that form a point cloud. This can be analyzed either by rendering an image from these coordinates, or by anal…