Can neural networks learn persistent homology features?

Montúfar, Guido; Otter, Nina; Wang, Yu Guang

doi:10.48550/arxiv.2011.14688

Cited by 6 publications

(7 citation statements)

References 10 publications

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“…Note, however, that the stability results also depend on the choice of filtration, persistence signature and metric [84]. Finally, we note that there has recently been a lot of effort in trying to train neural networks to learn what the best PH signature is for specific types of applications [19,36,59,74].…”

Section: E2 Discussion Of Ph Pipeline For Other Applicationsmentioning

confidence: 99%

On the effectiveness of persistent homology

Turkeš¹,

Montúfar²,

Otter³

2022

Preprint

Self Cite

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Persistent homology (PH) is one of the most popular methods in Topological Data Analysis. While PH has been used in many different types of applications, the reasons behind its success remain elusive. In particular, it is not known for which classes of problems it is most effective, or to what extent it can detect geometric or topological features. The goal of this work is to identify some types of problems on which PH performs well or even better than other methods in data analysis. We consider three fundamental shape-analysis tasks: the detection of the number of holes, curvature and convexity from 2D and 3D point clouds sampled from shapes. Experiments demonstrate that PH is successful in these tasks, outperforming several baselines, including PointNet, an architecture inspired precisely by the properties of point clouds. In addition, we observe that PH remains effective for limited computational resources and limited training data, as well as out-of-distribution test data, including various data transformations and noise.

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Section: E2 Discussion Of Ph Pipeline For Other Applicationsmentioning

confidence: 99%

On the effectiveness of persistent homology

Turkeš¹,

Montúfar²,

Otter³

2022

Preprint

Self Cite

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“…Topological Methods in ML With the marriage of homology and ML (Hensel, Moor, and Rieck 2021;Love et al 2021;Montúfar, Otter, and Wang 2020;Hofer, Kwitt, and Niethammer 2019), it did not take long for GNNs to meet their higher dimensional counterparts in the form of simplicial (Bodnar et al 2021b;Ebli, Defferrard, and Spreemann 2020;Bunch et al 2020), cell (Hajij, Istvan, and Zamzmi 2021;Bodnar et al 2021a), hypergraph (Feng et al 2019), and sheaf (Hansen and Gebhart 2020) neural networks. Most higher dimensional extensions of GNNs aim to operate on the full complex, and redefine the convolution operation in terms of the corresponding Laplacian operator.…”

Section: Related Workmentioning

confidence: 99%

Dist2Cycle: A Simplicial Neural Network for Homology Localization

Keros

Nanda

Subr

2022

AAAI

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Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher dimensional topological features of data, features to which graphs, encoding only pairwise relationships, remain oblivious. While attempts have been made to extend Graph Neural Networks (GNNs) to a simplicial complex setting, the methods do not inherently exploit, or reason about, the underlying topological structure of the network. We propose a graph convolutional model for learning functions parametrized by the k-homological features of simplicial complexes. By spectrally manipulating their combinatorial k-dimensional Hodge Laplacians, the proposed model enables learning topological features of the underlying simplicial complexes, specifically, the distance of each k-simplex from the nearest "optimal" k-th homology generator, effectively providing an alternative to homology localization.

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“…In [SCR + 20], the authors propose a convolutional neural network (CNN) architecture to estimate persistence images (see Section 2.2) computed on 2D-images. Similarly, in [MOW20], the authors provide an experimental overview of specific PD features (such as, e.g., their tropical coordinates [Kal19]) that can be learned using a CNN, when PDs are computed on top of 2D-images. On the other hand, RipsNet is designed to handle the (arguably harder) situation where input data are point clouds of arbitrary cardinality instead of 2D-images (i.e., vectors).…”

Section: Related Workmentioning

confidence: 99%

RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds

Surrel¹,

Hensel²,

Carrière³

et al. 2022

Preprint

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The use of topological descriptors in modern machine learning applications, such as Persistence Diagrams (PDs) arising from Topological Data Analysis (TDA), has shown great potential in various domains. However, their practical use in applications is often hindered by two major limitations: the computational complexity required to compute such descriptors exactly, and their sensitivity to even low-level proportions of outliers. In this work, we propose to bypass these two burdens in a data-driven setting by entrusting the estimation of (vectorization of) PDs built on top of point clouds to a neural network architecture that we call RipsNet. Once trained on a given data set, RipsNet can estimate topological descriptors on test data very efficiently with generalization capacity. Furthermore, we prove that RipsNet is robust to input perturbations in terms of the 1-Wasserstein distance, a major improvement over the standard computation of PDs that only enjoys Hausdorff stability, yielding RipsNet to substantially outperform exactly-computed PDs in noisy settings. We showcase the use of RipsNet on both synthetic and real-world data. Our open-source implementation is publicly available † and will be included in the Gudhi library.

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Can neural networks learn persistent homology features?

Cited by 6 publications

References 10 publications

On the effectiveness of persistent homology

On the effectiveness of persistent homology

Dist2Cycle: A Simplicial Neural Network for Homology Localization

RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds

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