Barcodes of Towers and a Streaming Algorithm for Persistent Homology

Kerber, Michael; Schreiber, Hannah

doi:10.1007/s00454-018-0030-0

Cited by 22 publications

(39 citation statements)

References 32 publications

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“…Our simplicial tower is connected by simplicial maps; there are (implemented) algorithms to compute the barcode of such towers (Dey et al 2014;Kerber and Schreiber 2017). It is also quite easy to adapt our tower construction to a streaming setting (Kerber and Schreiber 2017), where the output list of events is passed to an output stream instead of being stored in memory.…”

Section: Discussionmentioning

confidence: 99%

“…Recall that a simplicial map can be written as a composition of simplex inclusions and contractions of vertices (Dey et al 2014;Kerber and Schreiber 2017). That means, given the complex X α s , to describe the complex at the next scale α s+1 , it suffices to specify -which pairs of vertices in X α s map to the same image underg, and -which simplices in X α s+1 are included at scale X α s+1 .…”

Section: Algorithm Descriptionmentioning

confidence: 99%

“…However, to our knowledge, there has been no attempt to utilize them in computing approximations of filtrations. Also, while there are efficient methods to compute persistence for simplicial complexes connected with simplicial maps (Dey et al 2014;Kerber and Schreiber 2017), we are not aware of such counterparts for cubical complexes. Our contributions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Improved approximate rips filtrations with shifted integer lattices and cubical complexes

Choudhary

Kerber

Raghvendra

2021

J Appl. and Comput. Topology

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 $$\mathbb {R}^d$$ 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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Section: Discussionmentioning

confidence: 99%

Section: Algorithm Descriptionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Improved approximate rips filtrations with shifted integer lattices and cubical complexes

Choudhary

Kerber

Raghvendra

2021

J Appl. and Comput. Topology

Self Cite

View full text Add to dashboard Cite

show abstract

“…We will focus primarily on sequential approaches to persistent homology computation. Other, non-sequential approaches include the chunk algorithm [3], spectral sequence procedures [46,22], Morse-theoretic batch reduction [32,33,58,6,29,34,48,59,21], distributed algorithms [4,53,44], GPU acceleration [63,38], streaming [41], and homotopy collapse [9,20,8]. There are closely related techniques in matrix factorization and zigzag persistence [50,11,10].…”

Section: Related Literaturementioning

confidence: 99%

U-match factorization: sparse homological algebra, lazy cycle representatives, and dualities in persistent (co)homology

Hang¹,

Giusti²,

Ziegelmeier³

et al. 2021

Preprint

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Persistent homology is a leading tool in topological data analysis (TDA). Many problems in TDA can be solved via homological -and indeed, linear -algebra. However, matrices in this domain are typically large, with rows and columns numbered in billions. Low-rank approximation of such arrays typically destroys essential information; thus, new mathematical and computational paradigms are needed for very large, sparse matrices.We present the U-match matrix factorization scheme to address this challenge. U-match has two desirable features. First, it admits a compressed storage format that reduces the number of nonzero entries held in computer memory by one or more orders of magnitude over other common factorizations. Second, it permits direct solution of diverse problems in linear and homological algebra, without decompressing matrices stored in memory. These problems include look-up and retrieval of rows and columns; evaluation of birth/death times, and extraction of generators in persistent (co)homology; and, calculation of bases for boundary and cycle subspaces of filtered chain complexes. Such bases are key to unlocking a range of other topological techniques for use in TDA, and U-match factorization is designed to make such calculations broadly accessible to practitioners.As an application, we show that individual cycle representatives in persistent homology can be retrieved at time and memory costs orders of magnitude below current state of the art, via global duality. Moreover, the algebraic machinery needed to achieve this computation already exists in many modern solvers.

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“…However, to our knowledge, there has been no attempt to utilize them in computing approximations of filtrations. Also, while there are efficient methods to compute persistence for simplicial complexes connected with simplicial maps [12,21], we are not aware of such counterparts for cubical complexes.…”

Section: Introductionmentioning

confidence: 99%

Improved Approximate Rips Filtrations with Shifted Integer Lattices and Cubical Complexes

Choudhary¹,

Kerber²,

Raghvendra³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Barcodes of Towers and a Streaming Algorithm for Persistent Homology

Cited by 22 publications

References 32 publications

Improved approximate rips filtrations with shifted integer lattices and cubical complexes

Improved approximate rips filtrations with shifted integer lattices and cubical complexes

U-match factorization: sparse homological algebra, lazy cycle representatives, and dualities in persistent (co)homology

Improved Approximate Rips Filtrations with Shifted Integer Lattices and Cubical Complexes

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