Current survey of Clifford geometric algebra applications

Hitzer, Eckhard; Hildenbrand, Dietmar

doi:10.1002/mma.8316

Cited by 23 publications

(10 citation statements)

References 333 publications

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“…when feasible loops must be found between two given anchor residues, and to compute atomic coordinates from internal coordinates 25,26 . More specifically, Conformal Geometric Algebra (CGA, see Section 3.2) has seen application in the molecular distance problem 27,28,29,30 , which provides a suitable approach in selecting plausible protein conformation starting from NMR measurements affected by uncertainty, rather than in a learning-based approach.…”

Section: Related Workmentioning

confidence: 99%

Modeling orientational features via Geometric Algebra for 3D protein coordinates prediction

Pepe

Lasenby

2023

Preprint

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By protein structure prediction (PSP) we refer to the prediction of the 3-dimensional (3D) folding of a protein, known as tertiary structure, starting from its amino acid sequence, known as primary structure. The state-of-the-art in PSP is currently achieved by complex deep learning pipelines that require several input features extracted from amino acid sequences. It has been demonstrated that features that grasp the relative orientation of amino acids in space positively impacts the prediction accuracy of the 3D coordinates of atoms in the protein backbone. In this paper, we demonstrate the relevance of Geometric Algebra (GA) in instantiating orientational features for PSP problems. We do so by proposing two novel GA-based metrics which contain information on relative orientations of amino acid residues. We then employ these metrics as an additional input features to a Graph Transformer (GT) architecture to aid the prediction of the 3D coordinates of a protein, and compare them to classical angle-based metrics. We show how our GA features yield comparable results to angle maps in terms of accuracy of the predicted coordinates. This is despite being constructed from less initial information about the protein backbone. The features are also fewer and more informative, and can be (i) closely associated to protein secondary structures and (ii) more readily predicted compared to angle maps. We hence deduce that GA can be employed as a tool to simplify the modeling of protein structures and pack orientational information in a more natural and meaningful way.

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Section: Related Workmentioning

confidence: 99%

Modeling orientational features via Geometric Algebra for 3D protein coordinates prediction

Pepe

Lasenby

2023

Preprint

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“…In the past century, Clifford algebras have steadily gained popularity and found many applications across diverse fields of science in recent years 1,2 . An attractive feature of Clifford algebras is that they unify and generalize various branches of mathematics commonly applied in physics 3,4 .…”

Section: Introductionmentioning

confidence: 99%

Visualizing Stokes' theorem with Geometric Algebra

Klausen¹

2022

Preprint

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“…Here, we endeavor to extend this approach to the higher dimensional associative Clifford geometric algebra Cl (2,1), which plays an important role in geometry, physics, and computer science. [1][2][3][4][5][6][7][8] Namely, it is the physical algrebra of 2 + 1 space-time and the conformal geometric algebra Cl(1 + 1, 1) of one-dimensional Euclidean space R 1 . Our results may therefore be of special interest in the special theory of relativity and for the conformal geometry of a Euclidean line.…”

Section: Introductionmentioning

confidence: 99%

“…The importance of the polar representation of complex numbers and quaternions is widely known. Here, we endeavor to extend this approach to the higher dimensional associative Clifford geometric algebra

Cl\left(2,1\right)

, which plays an important role in geometry, physics, and computer science 1–8 . Namely, it is the physical algrebra of 2 + 1 space‐time and the conformal geometric algebra