書評 Computational Homology

By 坂上貴之. 2005

506 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/17573feca55a811144a3923760b9b5acefafb7a8

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In Computational Theory and Mathematics

In Mathematical Physics

In Signal Processing

Citations in the corpus

Betti numbers in multidimensional persistent homology are stable functions (2013, background): “These events can be encoded in the form of a parameterized version of the Betti numbe rs known in the literature aspersistent Betti numbers [21], a rank invariant [7], and, for the0th homology, asize function[23, 28, 34].”
Coverage in sensor networks via persistent homology (2007, background): “Increasingly, homology is a practical tools in applications which require computation of global structure: see the texts [10, 12] for an introduction, the latter containing several current applications in science and engineering.”
A roadmap for the computation of persistent homology (2015, background): “See [76] for a treatment of computational homology with cubical complexes rather than simplicial complexes and for a discussion of the relationship between simplicial and cubical homology.”
What are higher-order networks? (2021, methodology): “Tools for analyzing time series data and dynamical systems using topological methods have been developed by Harer, Perea, Robins, Mischaikow, Mukherjee among others (see [125,142,191,216,226] and references within).”
Persistent homology of complex networks (2008, background): “It should be remarked that these computations can be quite timeconsuming in spite of recent advances in computational techniques [8].”
Feature-based surface parameterization and texture mapping (2005, methodology): “This is achieved by treating R as a 2-complex (with boundary edges) and repeatedly performing elementary collapses [Kaczynski et al. 2004], in which a triangle with at least one boundary edge is removed from the complex along with one of the boundary edges.”
Topological Data Analysis (2012, methodology): “Currently, the witness complex is one of the few practical methods for topological analysis of large datasets.”
Finding the Homology of Submanifolds with High Confidence from Random Samples (2008, background): “Our results may also be of interest to researchers in computational geometry and topology who have considered the question of computing homology from simplicial complexes in the past (see [8, 14] for details and further references).”
Coverage and hole-detection in sensor networks via homology (2005, background): “This procedure can be generalized to higher dimensions as well [13].”
Barcodes: The persistent topology of data (2007, background): “The text [16] gives a comprehensive introduction to issues of and algorithms available for computing homology for realistic problems in application domains.”
On the Complexity of Neural Network Classifiers: A Comparison Between Shallow and Deep Architectures (2014, unknown): “All the above mentioned architectures have found in recent years a wide application to challenging problems in pattern recognition, such as, for example, in natural language processing [16], chemistry [17], image processing [18], [19], and web mining [20], [21].”
Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks (2021, background): “More broadly, the transformation could target a much wider output space such as cubical complexes (Kaczynski et al., 2004) (see Appendix E), or cell complexes (Hatcher, 2000), for which a message passing procedure has already been proposed (Hajij et al., 2020).”