Topology-Aware Surface Reconstruction for Point Clouds

Brüel-Gabrielsson, Rickard; Ganapathi-Subramanian, Vignesh; Škraba, Primož; Guibas, Leonidas J.

doi:10.48550/arxiv.1811.12543

Cited by 3 publications

(10 citation statements)

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“…Injectivity of the filter function over the vertices Graph learning [23] Any loss of supervised learning Shape matching [32] Bottleneck or Wasserstein distance Surface reconstruction [19] Sum of persistences Local correspondence between barcodes interval endpoints and vertices in the simplicial complex…”

Section: Parametrizations Of Lower Star Filtrationsmentioning

confidence: 99%

“…In [5,19], lower-star filtrations are parametrized for different purposes. In [19], functions on a grid are used to tackle the problem of surface reconstruction.…”

Section: Point Clouds Determining a Rips Filtrationmentioning

confidence: 99%

“…In [5,19], lower-star filtrations are parametrized for different purposes. In [19], functions on a grid are used to tackle the problem of surface reconstruction. These functions are sums of gaussians whose means and variances are parameters one wants to optimize according to an objective/loss that depends on the degree-1 persistent homology of the functions.…”

Section: Point Clouds Determining a Rips Filtrationmentioning

confidence: 99%

“…In both scenarios, Proposition 4.13 gives a formula to compute a differential of B p at θ from the barcode template pP p , U p q. The optimization pipelines mentioned in the introduction [12,19,20,23,32] apply this strategy to compute differentials.…”

Section: Differential Of the Barcode Valued Mapmentioning

confidence: 99%

“…Parametrizations of lower star filtrations are involved in [5,12,19,23,32], here we provide a common analysis of their differentiability.…”

Section: Lower Star Filtrationsmentioning

confidence: 99%

See 4 more Smart Citations

A Framework for Differential Calculus on Persistence Barcodes

Leygonie¹,

Oudot²,

Tillmann³

2019

Preprint

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We define notions of differentiability for maps from and to the space of persistence barcodes. Inspired by the theory of diffeological spaces, the proposed framework uses lifts to the space of ordered barcodes, from which derivatives can be computed. The two derived notions of differentiability (respectively from and to the space of barcodes) combine together naturally to produce a chain rule that enables the use of gradient descent for objective functions factoring through the space of barcodes. We illustrate the versatility of this framework by showing how it can be used to analyze the smoothness of various parametrized families of filtrations arising in topological data analysis.

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Section: Parametrizations Of Lower Star Filtrationsmentioning

confidence: 99%

“…In [5,19], lower-star filtrations are parametrized for different purposes. In [19], functions on a grid are used to tackle the problem of surface reconstruction.…”

Section: Point Clouds Determining a Rips Filtrationmentioning

confidence: 99%

Section: Point Clouds Determining a Rips Filtrationmentioning

confidence: 99%

Section: Differential Of the Barcode Valued Mapmentioning

confidence: 99%

“…Parametrizations of lower star filtrations are involved in [5,12,19,23,32], here we provide a common analysis of their differentiability.…”

Section: Lower Star Filtrationsmentioning

confidence: 99%

See 3 more Smart Citations

A Framework for Differential Calculus on Persistence Barcodes

Leygonie¹,

Oudot²,

Tillmann³

2019

Preprint

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Path Homologies of Deep Feedforward Networks

Chowdhury

Gebhart

Huntsman

et al. 2019

2019 18th IEEE International Conference on Machine Learning and Applications (ICMLA)

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We provide a characterization of two types of directed homology for fully-connected, feedforward neural network architectures. These exact characterizations of the directed homology structure of a neural network architecture are the first of their kind. We show that the directed flag homology of deep networks reduces to computing the simplicial homology of the underlying undirected graph, which is explicitly given by Euler characteristic computations. We also show that the path homology of these networks is non-trivial in higher dimensions and depends on the number and size of the layers within the network. These results provide a foundation for investigating homological differences between neural network architectures and their realized structure as implied by their parameters.

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Robust Persistence Diagrams using Reproducing Kernels

Vishwanath¹,

Fukumizu²,

Kuriki³

et al. 2020

Preprint

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Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, persistence diagrams are very sensitive to perturbations in the input space. In this work, we develop a framework for constructing robust persistence diagrams from superlevel filtrations of robust density estimators constructed using reproducing kernels. Using an analogue of the influence function on the space of persistence diagrams, we establish the proposed framework to be less sensitive to outliers. The robust persistence diagrams are shown to be consistent estimators in bottleneck distance, with the convergence rate controlled by the smoothness of the kernel-this in turn allows us to construct uniform confidence bands in the space of persistence diagrams. Finally, we demonstrate the superiority of the proposed approach on benchmark datasets.

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Topology-Aware Surface Reconstruction for Point Clouds

Cited by 3 publications

References 0 publications

A Framework for Differential Calculus on Persistence Barcodes

A Framework for Differential Calculus on Persistence Barcodes

Path Homologies of Deep Feedforward Networks

Robust Persistence Diagrams using Reproducing Kernels

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