Improved Topological Approximations by Digitization

Choudhary, Aruni; Kerber, Michael; Raghvendra, Sharath

doi:10.1137/1.9781611975482.166

Cited by 9 publications

(5 citation statements)

References 22 publications

(31 reference statements)

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“…Our approach is an instance of "lossless compression": the persistence module encoded in the output is isomorphic to the one of the input. In the singleparameter case, a popular research direction is the approximation of persistence diagrams, where the input is reduced in a way that the resulting module is provably close to the original one in terms of bottleneck distance (e.g., [33,17,12]). The prospects of this idea are unexplored in the multi-parameter setup.…”

Section: Discussionmentioning

confidence: 99%

Fast Minimal Presentations of Bi-graded Persistence Modules

Kerber¹,

Rolle²

2020

Preprint

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Multi-parameter persistent homology is a recent branch of topological data analysis. In this area, data sets are investigated through the lens of homology with respect to two or more scale parameters. The high computational cost of many algorithms calls for a preprocessing step to reduce the input size. In general, a minimal presentation is the smallest possible representation of a persistence module. Lesnick and Wright [28] proposed recently an algorithm (the LW-algorithm) for computing minimal presentations based on matrix reduction.In this work, we propose, implement and benchmark several improvements over the LW-algorithm. Most notably, we propose the use of priority queues to avoid extensive scanning of the matrix columns, which constitutes the computational bottleneck in the LWalgorithm, and we combine their algorithm with ideas from the multi-parameter chunk algorithm by Fugacci and Kerber [21]. Our extensive experiments show that our algorithm outperforms the LW-algorithm and computes the minimal presentation for data sets with millions of simplices within a few seconds. Our software is publicly available.

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Section: Discussionmentioning

confidence: 99%

Fast Minimal Presentations of Bi-graded Persistence Modules

Kerber¹,

Rolle²

2020

Preprint

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“…Further approximation techniques for Rips complexes (Dey et al 2014) and the closely relatedČech complexes (Botnan and Spreemann 2015;Cavanna et al 2015;Kerber and Sharathkumar 2013) have been derived subsequently, all with comparable size bounds. More recently, we constructed an approximation scheme (Choudhary et al 2019) for theČech filtrations of n points in R d that had size n 1 ε O(d) 2 O(d log d+dk) for the k-skeleton, improving the size bound from previous work.…”

Section: Introductionmentioning

confidence: 98%

Improved approximate rips filtrations with shifted integer lattices and cubical complexes

Choudhary

Kerber

Raghvendra

2021

J Appl. and Comput. Topology

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Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes is expensive because of a combinatorial explosion in the complex size. For n points in $$\mathbb {R}^d$$ R d , we present a scheme to construct a 2-approximation of the filtration of the Rips complex in the $$L_\infty $$ L ∞ -norm, which extends to a $$2d^{0.25}$$ 2 d 0.25 -approximation in the Euclidean case. The k-skeleton of the resulting approximation has a total size of $$n2^{O(d\log k +d)}$$ n 2 O ( d log k + d ) . The scheme is based on the integer lattice and simplicial complexes based on the barycentric subdivision of the d-cube. We extend our result to use cubical complexes in place of simplicial complexes by introducing cubical maps between complexes. We get the same approximation guarantee as the simplicial case, while reducing the total size of the approximation to only $$n2^{O(d)}$$ n 2 O ( d ) (cubical) cells. There are two novel techniques that we use in this paper. The first is the use of acyclic carriers for proving our approximation result. In our application, these are maps which relate the Rips complex and the approximation in a relatively simple manner and greatly reduce the complexity of showing the approximation guarantee. The second technique is what we refer to as scale balancing, which is a simple trick to improve the approximation ratio under certain conditions.

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“…Further approximation techniques for Rips complexes [12] and the closely related Čech complexes [1,7,22] have been derived subsequently, all with comparable size bounds. More recently, we constructed an approximation scheme [11] for the Čech filtrations of n points in R d that had size n…”

Section: Introductionmentioning

confidence: 99%

Improved Approximate Rips Filtrations with Shifted Integer Lattices and Cubical Complexes

Choudhary¹,

Kerber²,

Raghvendra³

2021

Preprint

Self Cite

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Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes is expensive because of a combinatorial explosion in the complex size. For n points in R d , we present a scheme to construct a 2-approximation of the filtration of the Rips complex in the L ∞ -norm, which extends to a 2d 0.25 -approximation in the Euclidean case. The k-skeleton of the resulting approximation has a total size of n2 O(d log k+d) . The scheme is based on the integer lattice and simplicial complexes based on the barycentric subdivision of the d-cube.We extend our result to use cubical complexes in place of simplicial complexes by introducing cubical maps between complexes. We get the same approximation guarantee as the simplicial case, while reducing the total size of the approximation to only n2 O(d) (cubical) cells.There are two novel techniques that we use in this paper. The first is the use of acyclic carriers for proving our approximation result. In our application, these are maps which relate the Rips complex and the approximation in a relatively simple manner and greatly reduce the complexity of showing the approximation guarantee. The second technique is what we refer to as scale balancing, which is a simple trick to improve the approximation ratio under certain conditions.

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Improved Topological Approximations by Digitization

Cited by 9 publications

References 22 publications

Fast Minimal Presentations of Bi-graded Persistence Modules

Fast Minimal Presentations of Bi-graded Persistence Modules

Improved approximate rips filtrations with shifted integer lattices and cubical complexes

Improved Approximate Rips Filtrations with Shifted Integer Lattices and Cubical Complexes

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