TopoGAN: A Topology-Aware Generative Adversarial Network

Wang, Fan; Liu, Huidong; Samaras, Dimitris; Chen, Chao

doi:10.1007/978-3-030-58580-8_8

Cited by 26 publications

(14 citation statements)

References 31 publications

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“…Based on the theory of algebraic topology (Munkres, 2018), persistent homology (Edelsbrunner et al, 2000;Edelsbrunner & Harer, 2010) extends the classical notion of homology, and can capture the topological structures (e.g., loops, connected components) of the input data in a robust (Cohen-Steiner et al, 2007) manner. It has already been combined with various deep learning methods including kernel machines (Reininghaus et al, 2015;Kusano et al, 2016;Carriere et al, 2017), convolutional neural networks (Hofer et al, 2017;Hu et al, 2019;Wang et al, 2020;Zheng et al, 2021), transformers (Zeng et al, 2021), connectivity loss (Chen et al, 2019;Hofer et al, 2019). and graph neural networks (Zhao et al, 2020;Yan et al, 2021;Zhao & Wang, 2019;Hofer et al, 2020;Carrière et al, 2020).…”

Section: Related Workmentioning

confidence: 99%

Neural Approximation of Graph Topological Features

Yan¹,

Ma²,

Gao³

et al. 2022

Preprint

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Persistent homology is a widely used theory in topological data analysis. In the context of graph learning, topological features based on persistent homology have been used to capture potentially high-order structural information so as to augment existing graph neural network methods. However, computing extended persistent homology summaries remains slow for large and dense graphs, especially since in learning applications one has to carry out this computation potentially many times. Inspired by recent success in neural algorithmic reasoning, we propose a novel learning method to compute extended persistence diagrams on graphs. The proposed neural network aims to simulate a specific algorithm and learns to compute extended persistence diagrams for new graphs efficiently. Experiments on approximating extended persistence diagrams and several downstream graph representation learning tasks demonstrate the effectiveness of our method. Our method is also efficient; on large and dense graphs, we accelerate the computation by nearly 100 times.

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Section: Related Workmentioning

confidence: 99%

Neural Approximation of Graph Topological Features

Yan¹,

Ma²,

Gao³

et al. 2022

Preprint

Self Cite

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“…1 (left). This has motivated previous work defining segmentation loss functions that encourage topologically consistent segmentations via persistent homology Hu et al (2019); Wang et al (2020a). Such losses compare the numbers of components and loops between ground truth and segmentation throughout a parameterized "filtration".…”

Section: Introductionmentioning

confidence: 98%

Quantifying Topology In Pancreatic Tubular Networks From Live Imaging 3D Microscopy

Arnavaz¹,

Krause²,

Zepf³

et al. 2021

Preprint

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Motivated by a challenging tubular network segmentation task, this paper tackles two commonly encountered problems in biomedical imaging: Topological consistency of the segmentation, and limited annotations. We propose a topological score which measures both topological and geometric consistency between the predicted and ground truth segmentations, applied for model selection and validation. We apply our topological score in three scenarios: i. a U-net ii. a U-net pretrained on an autoencoder, and iii. a semisupervised U-net architecture, which offers a straightforward approach to jointly training the network both as an autoencoder and a segmentation algorithm. This allows us to utilize un-annotated data for training a representation that generalizes across test data variability, in spite of our annotated training data having very limited variation. Our contributions are validated on a challenging segmentation task, locating tubular structures in the fetal pancreas from noisy live imaging confocal microscopy.

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“…By the stability theorem of PDs [25,71], close distances between shapes or functions on them imply close distances between their PDs; thus, computing diagram distances efficiently becomes important. It can help an increasing list of applications such as clustering [29,53,60], classification [17,56,72] and deep learning [76] that have found the use of topological persistence for analyzing data. The 1-Wasserstein (W 1 ) distance is a common distance to compare persistence diagrams; Hera [50] is a widely used open source software for this.…”

Section: Introductionmentioning

confidence: 99%

Approximating 1-Wasserstein Distance between Persistence Diagrams by Graph Sparsification

Dey¹,

Zhang²

2021

Preprint

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Persistence diagrams (PD)s play a central role in topological data analysis. This analysis requires computing distances among such diagrams such as the 1-Wasserstein distance. Accurate computation of these PD distances for large data sets that render large diagrams may not scale appropriately with the existing methods. The main source of difficulty ensues from the size of the bipartite graph on which a matching needs to be computed for determining these PD distances. We address this problem by making several algorithmic and computational observations in order to obtain an approximation. First, taking advantage of the proximity of PD points, we condense them thereby decreasing the number of nodes in the graph for computation. The increase in point multiplicities is addressed by reducing the matching problem to a min-cost flow problem on a transshipment network. Second, we use Well Separated Pair Decomposition to sparsify the graph to a size that is linear in the number of points. Both node and arc sparsifications contribute to the approximation factor where we leverage a lower bound given by the Relaxed Word Mover's distance. Third, we eliminate bottlenecks during the sparsification procedure by introducing parallelism. Fourth, we develop an open source software called 1 PDoptFlow based on our algorithm, exploiting parallelism by GPU and multicore. We perform extensive experiments and show that the actual empirical error is very low. We also show that we can achieve high performance at low guaranteed relative errors, improving upon the state of the arts.

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TopoGAN: A Topology-Aware Generative Adversarial Network

Cited by 26 publications

References 31 publications

Neural Approximation of Graph Topological Features

Neural Approximation of Graph Topological Features

Quantifying Topology In Pancreatic Tubular Networks From Live Imaging 3D Microscopy

Approximating 1-Wasserstein Distance between Persistence Diagrams by Graph Sparsification

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