Pub Date : 2021-08-17DOI: 10.1007/s41468-022-00100-x
Jacob Leygonie, David Beers
{"title":"Fiber of persistent homology on morse functions","authors":"Jacob Leygonie, David Beers","doi":"10.1007/s41468-022-00100-x","DOIUrl":"https://doi.org/10.1007/s41468-022-00100-x","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"7 1","pages":"89 - 102"},"PeriodicalIF":0.0,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47085147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-24DOI: 10.1007/s41468-021-00074-2
Michael J. Catanzaro, V. Chernyak, John R. Klein
{"title":"Correction to: A higher Boltzmann distribution","authors":"Michael J. Catanzaro, V. Chernyak, John R. Klein","doi":"10.1007/s41468-021-00074-2","DOIUrl":"https://doi.org/10.1007/s41468-021-00074-2","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"5 1","pages":"531 - 532"},"PeriodicalIF":0.0,"publicationDate":"2021-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s41468-021-00074-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47821214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-30DOI: 10.1007/s41468-022-00089-3
J. Curry, Haibin Hang, W. Mio, Tom Needham, Osman Berat Okutan
{"title":"Decorated merge trees for persistent topology","authors":"J. Curry, Haibin Hang, W. Mio, Tom Needham, Osman Berat Okutan","doi":"10.1007/s41468-022-00089-3","DOIUrl":"https://doi.org/10.1007/s41468-022-00089-3","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"6 1","pages":"371 - 428"},"PeriodicalIF":0.0,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47446113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.1007/s41468-021-00083-1
Michal Adamaszek, Henry Adams
{"title":"On Vietoris–Rips complexes of hypercube graphs","authors":"Michal Adamaszek, Henry Adams","doi":"10.1007/s41468-021-00083-1","DOIUrl":"https://doi.org/10.1007/s41468-021-00083-1","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"6 1","pages":"177 - 192"},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46961623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-12DOI: 10.1007/s41468-020-00064-w
David Recio-Mitter
{"title":"Geodesic complexity of motion planning","authors":"David Recio-Mitter","doi":"10.1007/s41468-020-00064-w","DOIUrl":"https://doi.org/10.1007/s41468-020-00064-w","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"79 1","pages":"141 - 178"},"PeriodicalIF":0.0,"publicationDate":"2021-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s41468-020-00064-w","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"53194030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01Epub Date: 2021-05-15DOI: 10.1007/s41468-021-00072-4
Aruni Choudhary, Michael Kerber, Sharath Raghvendra
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 , we present a scheme to construct a 2-approximation of the filtration of the Rips complex in the -norm, which extends to a -approximation in the Euclidean case. The k-skeleton of the resulting approximation has a total size of . 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 (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.
撕裂复形是分析度量空间拓扑特征的重要结构。不幸的是，由于复合尺寸的组合爆炸，生成这些复合物是昂贵的。对于R d中的n个点，我们给出了在L∞范数下构造Rips复滤的2-近似的方案，该方案扩展到欧几里得情况下的2- d 0.25 -近似。所得到的近似的k骨架的总大小为n2o (d log k + d)。该方案基于整数点阵和基于d-立方体质心细分的简单复形。通过引入复形之间的三次映射，我们扩展了用三次复形代替简单复形的结果。我们得到了与简单情况相同的近似保证，同时将近似的总大小减小到只有n 2o (d)(立方)单元。我们在本文中使用了两种新颖的技术。第一个是使用无环载流子来证明我们的近似结果。在我们的应用中，这些映射以一种相对简单的方式将Rips复合体和近似值联系起来，并大大降低了显示近似值保证的复杂性。第二种技术是我们所说的尺度平衡，这是一种在特定条件下提高近似比的简单技巧。
{"title":"Improved approximate rips filtrations with shifted integer lattices and cubical complexes.","authors":"Aruni Choudhary, Michael Kerber, Sharath Raghvendra","doi":"10.1007/s41468-021-00072-4","DOIUrl":"https://doi.org/10.1007/s41468-021-00072-4","url":null,"abstract":"<p><p>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 <i>n</i> points in <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> , we present a scheme to construct a 2-approximation of the filtration of the Rips complex in the <math><msub><mi>L</mi> <mi>∞</mi></msub> </math> -norm, which extends to a <math><mrow><mn>2</mn> <msup><mi>d</mi> <mrow><mn>0.25</mn></mrow> </msup> </mrow> </math> -approximation in the Euclidean case. The <i>k</i>-skeleton of the resulting approximation has a total size of <math><mrow><mi>n</mi> <msup><mn>2</mn> <mrow><mi>O</mi> <mo>(</mo> <mi>d</mi> <mo>log</mo> <mi>k</mi> <mo>+</mo> <mi>d</mi> <mo>)</mo></mrow> </msup> </mrow> </math> . The scheme is based on the integer lattice and simplicial complexes based on the barycentric subdivision of the <i>d</i>-cube. We extend our result to use cubical complexes in place of simplicial complexes by introducing <i>cubical maps</i> between complexes. We get the same approximation guarantee as the simplicial case, while reducing the total size of the approximation to only <math><mrow><mi>n</mi> <msup><mn>2</mn> <mrow><mi>O</mi> <mo>(</mo> <mi>d</mi> <mo>)</mo></mrow> </msup> </mrow> </math> (cubical) cells. There are two novel techniques that we use in this paper. The first is the use of <i>acyclic carriers</i> 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 <i>scale balancing</i>, which is a simple trick to improve the approximation ratio under certain conditions.</p>","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"5 3","pages":"425-458"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s41468-021-00072-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39580559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-18DOI: 10.1007/s41468-022-00091-9
Peter Bubenik, Alex Elchesen
{"title":"Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces","authors":"Peter Bubenik, Alex Elchesen","doi":"10.1007/s41468-022-00091-9","DOIUrl":"https://doi.org/10.1007/s41468-022-00091-9","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"6 1","pages":"429 - 474"},"PeriodicalIF":0.0,"publicationDate":"2020-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46803814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-17DOI: 10.1007/s41468-023-00121-0
P. Schapira
{"title":"Constructible sheaves and functions up to infinity","authors":"P. Schapira","doi":"10.1007/s41468-023-00121-0","DOIUrl":"https://doi.org/10.1007/s41468-023-00121-0","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"7 1","pages":"707 - 739"},"PeriodicalIF":0.0,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42927274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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