Optimal evaluation of symmetry-adapted n-correlations via recursive contraction of sparse symmetric tensors
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Illia Kaliuzhnyi
Abstract
We present a comprehensive analysis of an algorithm for evaluating high-dimensional polynomials that are invariant (or equi-variant) under permutations and rotations. This task arises in the evaluation of linear models as well as equivariant neural network models of many-particle systems. The theoretical bottleneck is the contraction of a high-dimensional symmetric and sparse tensor with a specific sparsity pattern that is directly related to the symmetries imposed on the polynomial. The sparsity of this tensor makes it challenging to construct a highly efficient evaluation scheme. Bachmayr et al. (“Polynomial approximation of symmetric functions,” Math. Comp., vol. 93, pp. 811–839, 2024) and Lysogorskiy et al. (“Performant implementation of the atomic cluster expansion (pace): application to copper and silicon,” npj Comput. Mater., vol. 7, Art. no. 97, 2021) introduced a recursive evaluation strategy that relied on a number of heuristics, but performed well in tests. In the present work, we propose an explicit construction of such a recursive evaluation strategy and show that it is in fact optimal in the limit of infinite polynomial degree.
Funding source: NSERC Discovery Grant
Award Identifier / Grant number: GR019381
Funding source: NFRF Exploration Grant
Award Identifier / Grant number: GR022937
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Research ethics: Not applicable.
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Informed consent: Not applicable.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Use of Large Language Models, AI and Machine Learning Tools: None declared.
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Conflict of interest: The authors state no conflict of interest.
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Research funding: This work was supported by NSERC Discovery Grant GR019381; NFRF Exploration Grant GR022937.
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Data availability: Not applicable.
References
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© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Stabilized weighted reduced order methods for parametrized advection-dominated optimal control problems governed by partial differential equations with random inputs
- Augmenting the grad-div stabilization for Taylor–Hood finite elements with a vorticity stabilization
- Obtaining higher-order Galerkin accuracy when the boundary is polygonally approximated
- Optimal evaluation of symmetry-adapted n-correlations via recursive contraction of sparse symmetric tensors
Articles in the same Issue
- Frontmatter
- Stabilized weighted reduced order methods for parametrized advection-dominated optimal control problems governed by partial differential equations with random inputs
- Augmenting the grad-div stabilization for Taylor–Hood finite elements with a vorticity stabilization
- Obtaining higher-order Galerkin accuracy when the boundary is polygonally approximated
- Optimal evaluation of symmetry-adapted n-correlations via recursive contraction of sparse symmetric tensors
