Mapper-type algorithms for complex data and relations

Dlotko, Pawel; Gurnari, Davide; Sazdanovic, Radmila

doi:10.48550/arxiv.2109.00831

Cited by 2 publications

(2 citation statements)

References 18 publications

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“…Nevertheless, due to supersymmetric localization, there is a classical solution of the supersymmetric equations with the appropriate quantum numbers; what we cannot say is whether this solution is unique or if there are others which complicate the limit. 21 As well, [35,36] study many of the same invariants we do from dimensionality reduction and topological data analysis perspectives.…”

Section: Methodsmentioning

confidence: 99%

Learning knot invariants across dimensions

Craven¹,

Hughes²,

Jejjala³

et al. 2023

SciPost Phys.

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We use deep neural networks to machine learn correlations between knot invariants in various dimensions. The three-dimensional invariant of interest is the Jones polynomial J(q)J(q), and the four-dimensional invariants are the Khovanov polynomial \text{Kh}(q,t)Kh(q,t), smooth slice genus gg, and Rasmussen’s ss-invariant. We find that a two-layer feed-forward neural network can predict ss from \text{Kh}(q,-q^{-4})Kh(q,−q−4) with greater than 99%99% accuracy. A theoretical explanation for this performance exists in knot theory via the now disproven knight move conjecture, which is obeyed by all knots in our dataset. More surprisingly, we find similar performance for the prediction of ss from \text{Kh}(q,-q^{-2})Kh(q,−q−2), which suggests a novel relationship between the Khovanov and Lee homology theories of a knot. The network predicts gg from \text{Kh}(q,t)Kh(q,t) with similarly high accuracy, and we discuss the extent to which the machine is learning ss as opposed to gg, since there is a general inequality |s| ≤2g|s|≤2g. The Jones polynomial, as a three-dimensional invariant, is not obviously related to ss or gg, but the network achieves greater than 95%95% accuracy in predicting either from J(q)J(q). Moreover, similar accuracy can be achieved by evaluating J(q)J(q) at roots of unity. This suggests a relationship with SU(2)SU(2) Chern—Simons theory, and we review the gauge theory construction of Khovanov homology which may be relevant for explaining the network’s performance.

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Section: Methodsmentioning

confidence: 99%

Learning knot invariants across dimensions

Craven¹,

Hughes²,

Jejjala³

et al. 2023

SciPost Phys.

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“…To our knowledge, the only other work linking PH and low-dimensional topology is new and considers only closed knots [ 32 ]. In a different direction, Mapper [ 33 ], another tool from computational topology, has been applied to the abstract collection of closed knots [ 34 ] parametrized by the values taken by polynomial invariants rather than the geometry of any specific embeddings. We propose and implement a computationally feasible PH pipeline for studying knotted proteins, at a global (full sequence) and local (substructural) scale, which does not rely on complex and computationally expensive knot invariants and sub-chain analysis [ 5 , 9 , 13 , 16 ].…”

Section: Introductionmentioning

confidence: 99%

Homology of homologous knotted proteins

Benjamin

Mukta

Moryoussef

et al. 2023

J. R. Soc. Interface.

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Quantification and classification of protein structures, such as knotted proteins, often requires noise-free and complete data. Here, we develop a mathematical pipeline that systematically analyses protein structures. We showcase this geometric framework on proteins forming open-ended trefoil knots, and we demonstrate that the mathematical tool, persistent homology, faithfully represents their structural homology. This topological pipeline identifies important geometric features of protein entanglement and clusters the space of trefoil proteins according to their depth. Persistence landscapes quantify the topological difference between a family of knotted and unknotted proteins in the same structural homology class. This difference is localized and interpreted geometrically with recent advancements in systematic computation of homology generators. The topological and geometric quantification we find is robust to noisy input data, which demonstrates the potential of this approach in contexts where standard knot theoretic tools fail.

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Mapper-type algorithms for complex data and relations

Cited by 2 publications

References 18 publications

Learning knot invariants across dimensions

Learning knot invariants across dimensions

Homology of homologous knotted proteins

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