Journal of applied and computational topology最新文献

{"title":"Morse theoretic signal compression and reconstruction on chain complexes.","authors":"Stefania Ebli, Celia Hacker, Kelly Maggs","doi":"10.1007/s41468-024-00191-8","DOIUrl":"https://doi.org/10.1007/s41468-024-00191-8","url":null,"abstract":"<p><p>At the intersection of Topological Data Analysis (TDA) and machine learning, the field of cellular signal processing has advanced rapidly in recent years. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian, and the resultant Hodge decomposition. Meanwhile, discrete Morse theory has been widely used to speed up computations by reducing the size of complexes while preserving their global topological properties. In this paper, we provide an approach to signal compression and reconstruction on chain complexes that leverages the tools of algebraic discrete Morse theory. The main goal is to reduce and reconstruct a based chain complex together with a set of signals on its cells via deformation retracts, preserving as much as possible the global topological structure of both the complex and the signals. We first prove that any deformation retract of real degree-wise finite-dimensional based chain complexes is equivalent to a Morse matching. We will then study how the signal changes under particular types of Morse matchings, showing its reconstruction error is trivial on specific components of the Hodge decomposition. Furthermore, we provide an algorithm to compute Morse matchings with minimal reconstruction error.</p>","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"8 8","pages":"2285-2326"},"PeriodicalIF":0.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11541343/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142634048","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}

{"title":"Metric geometry of spaces of persistence diagrams.","authors":"Mauricio Che, Fernando Galaz-García, Luis Guijarro, Ingrid Amaranta Membrillo Solis","doi":"10.1007/s41468-024-00189-2","DOIUrl":"https://doi.org/10.1007/s41468-024-00189-2","url":null,"abstract":"<p><p>Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors <math><msub><mi>D</mi> <mi>p</mi></msub> </math> , <math><mrow><mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi></mrow> </math> , that assign, to each metric pair (<i>X</i>, <i>A</i>), a pointed metric space <math> <mrow><msub><mi>D</mi> <mi>p</mi></msub> <mrow><mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo>)</mo></mrow> </mrow> </math> . Moreover, we show that <math><msub><mi>D</mi> <mi>∞</mi></msub> </math> is sequentially continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that <math><msub><mi>D</mi> <mi>p</mi></msub> </math> preserves several useful metric properties, such as completeness and separability, for <math><mrow><mi>p</mi> <mo>∈</mo> <mo>[</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo>)</mo></mrow> </math> , and geodesicity and non-negative curvature in the sense of Alexandrov, for <math><mrow><mi>p</mi> <mo>=</mo> <mn>2</mn></mrow> </math> . For the latter case, we describe the metric of the space of directions at the empty diagram. We also show that the Fréchet mean set of a Borel probability measure on <math> <mrow><msub><mi>D</mi> <mi>p</mi></msub> <mrow><mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo>)</mo></mrow> </mrow> </math> , <math><mrow><mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi></mrow> </math> , with finite second moment and compact support is non-empty. As an application of our geometric framework, we prove that the space of Euclidean persistence diagrams, <math> <mrow><msub><mi>D</mi> <mi>p</mi></msub> <mrow><mo>(</mo> <msup><mrow><mi>R</mi></mrow> <mrow><mn>2</mn> <mi>n</mi></mrow> </msup> <mo>,</mo> <msub><mi>Δ</mi> <mi>n</mi></msub> <mo>)</mo></mrow> </mrow> </math> , <math><mrow><mn>1</mn> <mo>≤</mo> <mi>n</mi></mrow> </math> and <math><mrow><mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi></mrow> </math> , has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad-Nagata dimensions.</p>","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"8 8","pages":"2197-2246"},"PeriodicalIF":0.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11541355/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142634046","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}

{"title":"Geometric characterization of the persistence of 1D maps.","authors":"Ranita Biswas, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, Morteza Saghafian","doi":"10.1007/s41468-023-00126-9","DOIUrl":"10.1007/s41468-023-00126-9","url":null,"abstract":"<p><p>We characterize critical points of 1-dimensional maps paired in persistent homology geometrically and this way get elementary proofs of theorems about the symmetry of persistence diagrams and the variation of such maps. In particular, we identify branching points and endpoints of networks as the sole source of asymmetry and relate the cycle basis in persistent homology with a version of the stable marriage problem. Our analysis provides the foundations of fast algorithms for maintaining a collection of sorted lists together with its persistence diagram.</p>","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":" ","pages":"1101-1119"},"PeriodicalIF":0.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11639680/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44570458","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}

{"title":"Defining logical obstruction with fixpoints in epistemic logic","authors":"Susumu Nishimura","doi":"10.1007/s41468-023-00151-8","DOIUrl":"https://doi.org/10.1007/s41468-023-00151-8","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"79 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134900728","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}

{"title":"Generalization of graph network inferences in higher-order graphical models","authors":"Yicheng Fei, Xaq Pitkow","doi":"10.1007/s41468-023-00147-4","DOIUrl":"https://doi.org/10.1007/s41468-023-00147-4","url":null,"abstract":"Abstract Probabilistic graphical models provide a powerful tool to describe complex statistical structure, with many real-world applications in science and engineering from controlling robotic arms to understanding neuronal computations. A major challenge for these graphical models is that inferences such as marginalization are intractable for general graphs. These inferences are often approximated by a distributed message-passing algorithm such as Belief Propagation, which does not always perform well on graphs with cycles, nor can it always be easily specified for complex continuous probability distributions. Such difficulties arise frequently in expressive graphical models that include intractable higher-order interactions. In this paper we define the Recurrent Factor Graph Neural Network (RF-GNN) to achieve fast approximate inference on graphical models that involve many-variable interactions. Experimental results on several families of graphical models demonstrate the out-of-distribution generalization capability of our method to different sized graphs, and indicate the domain in which our method outperforms Belief Propagation (BP). Moreover, we test the RF-GNN on a real-world Low-Density Parity-Check dataset as a benchmark along with other baseline models including BP variants and other GNN methods. Overall we find that RF-GNNs outperform other methods under high noise levels.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"61 16","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136283777","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}

{"title":"The topology of randomized symmetry-breaking distributed computing","authors":"Pierre Fraigniaud, Ran Gelles, Zvi Lotker","doi":"10.1007/s41468-023-00150-9","DOIUrl":"https://doi.org/10.1007/s41468-023-00150-9","url":null,"abstract":"Studying distributed computing through the lens of algebraic topology has been the source of many significant breakthroughs during the last 2 decades, especially in the design of lower bounds or impossibility results. Despite hundred of results considering deterministic algorithms, none apply to randomized algorithms. This paper aims at studying randomized synchronous distributed computing through the lens of algebraic topology. We do so by studying the wide class of (input-free) symmetry-breaking tasks, e.g., leader election, in synchronous fault-free anonymous systems. We design a topological framework, which allows analyzing such tasks and determining their solvability. The pivotal technical observation is that, unlike in deterministic algorithm, where solvability means that the topological complex describing the protocol can be globally mapped into an output protocol, in our framework the solvability is determined “locally”, i.e., for each simplex of the protocol complex individually, without requiring any global consistency. As an interesting application, we derive necessary and sufficient conditions for solving leader election in shared-memory and message-passing models in which there might be correlations between the randomness provided to the nodes. We find that solvability of leader election relates to the number of parties that possess correlated randomness, either directly or via their greatest common divisor, depending on the specific communication model.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"9 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134993015","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}

{"title":"Simplicial branching random walks","authors":"Ron Rosenthal","doi":"10.1007/s41468-023-00148-3","DOIUrl":"https://doi.org/10.1007/s41468-023-00148-3","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"114 30","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135138044","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}

{"title":"Rigorous computation in dynamics based on topological methods for multivector fields","authors":"Donald Woukeng, Damian Sadowski, Jakub Leśkiewicz, Michał Lipiński, Tomasz Kapela","doi":"10.1007/s41468-023-00149-2","DOIUrl":"https://doi.org/10.1007/s41468-023-00149-2","url":null,"abstract":"Abstract Motivated by the theoretical results of Mrozek et al. (Commun Nonlinear Sci Numer Simul 108:106–226, 2022) we present an algorithmic construction of a transversal cellular decomposition for a planar ODE. We then use the associated combinatorial multivector field to algorithmically detect the existence of an isolated invariant set with the Conley index of a periodic orbit and admitting a combinatorial Poincaré section. This construction combined with the theoretical results of Mrozek et al. (2022) leads to a method for automatized computer assisted proofs of the existence of periodic solutions in ODE’s.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"28 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135774489","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}

{"title":"Random simple-homotopy theory","authors":"Bruno Benedetti, Crystal Lai, Davide Lofano, Frank H. Lutz","doi":"10.1007/s41468-023-00139-4","DOIUrl":"https://doi.org/10.1007/s41468-023-00139-4","url":null,"abstract":"Abstract We implement an algorithm RSHT (random simple-homotopy) to study the simple-homotopy types of simplicial complexes, with a particular focus on contractible spaces and on finding substructures in higher-dimensional complexes. The algorithm combines elementary simplicial collapses with pure elementary expansions . For triangulated d -manifolds with $$dle 6$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> , we show that RSHT reduces to (random) bistellar flips. Among the many examples on which we test RSHT, we describe an explicit 15-vertex triangulation of the Abalone, and more generally, $$(14k+1)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>14</mml:mn> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -vertex triangulations of a new series of Bing’s houses with k rooms, $$kge 3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , which all can be deformed to a point using only six pure elementary expansions.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136157610","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}

{"title":"Yarn ball knots and faster computations","authors":"Dror Bar-Natan, Itai Bar-Natan, Iva Halacheva, Nancy Scherich","doi":"10.1007/s41468-023-00144-7","DOIUrl":"https://doi.org/10.1007/s41468-023-00144-7","url":null,"abstract":"Abstract We make use of the 3D nature of knots and links to find savings in computational complexity when computing knot invariants such as the linking number and, in general, most finite type invariants. These savings are achieved in comparison with the 2D approach to knots using knot diagrams.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"19 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136157611","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}