Homologies of digraphs and Künneth formulas

Grigor’yan, Alexander; Muranov, Yu. V.; Yau, Shing–Tung; yan, Alexander Grigor

doi:10.4310/cag.2017.v25.n5.a4

Cited by 48 publications

(58 citation statements)

References 17 publications

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“…Prior to providing the proof of Theorem 1, we digress briefly to highlight an interesting connection. The theory of path homology admits Künneth formulas for various digraph constructions [32], and one might expect that the layered construction of feedforward neural networks would be amenable to applying such formulas. Indeed, when restricted to feedforward network architectures having two layers, the Künneth formula for join applies to give the result in Theorem 1.…”

Section: Path Homology Of Mlpsmentioning

confidence: 99%

Path Homologies of Deep Feedforward Networks

Chowdhury

Gebhart

Huntsman

et al. 2019

2019 18th IEEE International Conference on Machine Learning and Applications (ICMLA)

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We provide a characterization of two types of directed homology for fully-connected, feedforward neural network architectures. These exact characterizations of the directed homology structure of a neural network architecture are the first of their kind. We show that the directed flag homology of deep networks reduces to computing the simplicial homology of the underlying undirected graph, which is explicitly given by Euler characteristic computations. We also show that the path homology of these networks is non-trivial in higher dimensions and depends on the number and size of the layers within the network. These results provide a foundation for investigating homological differences between neural network architectures and their realized structure as implied by their parameters.

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Section: Path Homology Of Mlpsmentioning

confidence: 99%

Path Homologies of Deep Feedforward Networks

Chowdhury

Gebhart

Huntsman

et al. 2019

2019 18th IEEE International Conference on Machine Learning and Applications (ICMLA)

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“…In what follows, we summarize and condense some concepts that appeared in [GLMY12], and attempt to preserve the original notation wherever possible. Definition 1.…”

Section: Background On Digraphs and Path Homologymentioning

confidence: 99%

“…However, as noted in [GLMY12], functorial properties such as the Künneth formula may fail for this construction-a cycle graph on four nodes has nontrivial 1-dimensional homology, but the Cartesian product of two such cycle graphs has trivial 2-dimensional homology in this clique complex formulation. On the other hand, the path homology construction of [GLMY12] satisfies a Künneth formula. We do not focus on comparing the different constructions of homology for graphs, but we list some desirable properties of path homology:…”

Section: Introductionmentioning

confidence: 99%

“…For example, one approach would be to define the clique complex of the underlying undirected graph, and then consider the homology of the clique complex [Iva94]. However, as noted in [GLMY12], functorial properties such as the Künneth formula may fail for this construction-a cycle graph on four nodes has nontrivial 1-dimensional homology, but the Cartesian product of two such cycle graphs has trivial 2-dimensional homology in this clique complex formulation. On the other hand, the path homology construction of [GLMY12] satisfies a Künneth formula.…”

Section: Introductionmentioning

confidence: 99%

“…In a series of papers released between 2012 and 2014, A. Grigor'yan, Y. Lin, Y. Muranov, and S.T. Yau formalized a notion of homology on digraphs called path homology [GLMY12], as well as a homotopy theory for digraphs that is compatible with path homology [GLMY14] and the homotopy theory for undirected graphs proposed in [BKLW01,BBLL06]. This notion of path homology is the central object of study in our work, and our contribution is to study the notion of persistent homology that arises from this construction.…”

Section: Introductionmentioning

confidence: 99%

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Persistent Path Homology of Directed Networks

Chowdhury

Mémoli

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Abstract. While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. We study a construction of homology of digraphs due to Grigor'yan, Lin, Muranov and Yau, and extend this construction to the persistent framework. The result, which we call persistent path homology, can provide information about the digraph structure of a directed network at varying resolutions. Moreover, this method encodes a rich level of detail about the asymmetric structure of the input directed network. We test our method on both simulated and real-world directed networks and conjecture some of its characteristics.

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