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Efficient generation and explicit dimensionality of Lie group-equivariant and permutation-invariant bases

Eloïse Barthelemy, Geneviève Dusson, Camille Hernandez, Liwei Zhang

Abstract

In this article, we propose a practical construction of Lie group-equivariant and permutation-invariant functions of $N$ variables from the knowledge of a one-particle basis that is stable with respect to the group action. The construction is generic for any linear Lie group and relies on building a matrix constructed from the Lie algebra whose kernel is spanned by a group-equivariant and permutation-invariant basis. In particular, this construction does not require the knowledge of Clebsch--Gordan coefficients and instead directly builds generalized Clebsch--Gordan coefficients. For specific groups such as $SO(3)$ and $SU(2)$, we exploit the Lie algebra structure to simplify the matrix, which then allows us to derive an explicit formula for the exact dimension of the group-equivariant and permutation-invariant space. Numerical simulations are provided to show that the proposed method scales linearly instead of exponentially for existing methods in the literature. We also show that for large values of $N$, the number of rotation-equivariant and permutation-invariant basis functions is of a comparable order as the number of permutation-invariant basis functions, while pre-asymptotically, a large gain can be achieved by explicitly enforcing rotation-equivariance on top of permutation-invariance.

Efficient generation and explicit dimensionality of Lie group-equivariant and permutation-invariant bases

Abstract

In this article, we propose a practical construction of Lie group-equivariant and permutation-invariant functions of variables from the knowledge of a one-particle basis that is stable with respect to the group action. The construction is generic for any linear Lie group and relies on building a matrix constructed from the Lie algebra whose kernel is spanned by a group-equivariant and permutation-invariant basis. In particular, this construction does not require the knowledge of Clebsch--Gordan coefficients and instead directly builds generalized Clebsch--Gordan coefficients. For specific groups such as and , we exploit the Lie algebra structure to simplify the matrix, which then allows us to derive an explicit formula for the exact dimension of the group-equivariant and permutation-invariant space. Numerical simulations are provided to show that the proposed method scales linearly instead of exponentially for existing methods in the literature. We also show that for large values of , the number of rotation-equivariant and permutation-invariant basis functions is of a comparable order as the number of permutation-invariant basis functions, while pre-asymptotically, a large gain can be achieved by explicitly enforcing rotation-equivariance on top of permutation-invariance.

Paper Structure

This paper contains 41 sections, 32 theorems, 260 equations, 9 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Generating equivariant bases using the Lie Algebra
  3. Preliminary
  4. Group-equivariant basis
  5. Group-equivariant and permutation invariant basis
  6. Recursive construction of GE and GE-PI bases
  7. Application to some rotation groups
  8. Representations and Lie algebra
  9. One-particle compatible bases
  10. Wigner-D matrices
  11. Spherical harmonics
  12. One-particle basis on $\mathbb{R}^3$
  13. One-particle basis on higher-dimensional spaces
  14. Construction of multi-variable GE bases
  15. Matrices ${M}_1$ and ${M}_3$
  16. ...and 26 more sections

Key Result

Proposition 2.4

Let ${\bm l}\in\mathcal{L}^{N}$, $L\in\mathcal{L}$, and let where for ${d} = 1,\ldots, N_{\rm{dim}}$, the elements of the submatrices $M_{d}$ are defined for $({\bm m},k),\; ({\bm m}',k') \in \mathcal{M}_{\bm l}\times\mathcal{M}_L$ as Then, a basis of the GE space $V^{{\bm l},L}$ is given by where ${\bm c}^{{\bm l},L}_i = \{c_{{\bm m},k,i}^{{\bm l},L}\}_{{\bm m}\in\mathcal{M}_{\bm l},k\in\mathc

Figures (9)

  • Figure 1: Block structure of matrix ${M}_2$: (a) ${M}_2$: original derivation; (b) $\widetilde{{M}}_2$: Row reordering of the ${M}_2$ to enable a fast kernel solver; (c) ${M}^{\textup{up}}$: upper half of $\widetilde{{M}}_2$; the final matrix for which we find the kernel. In the plots, the red, green and blue boxes correspond to the matrix in the cases $L<\sum{\bm l},\; L=\sum{\bm l}$, and $L>\sum{\bm l}$, respectively. All the blocks left blank are zeros.
  • Figure 2: Breakdown of construction time (ms) of the GE-PI bases versus the number of classes in the matrix ${{M}}^{\textup{up}}$\ref{['eq:M2-PI']}, or the product of the number of classes and the number of basis functions.
  • Figure 3: Run time for constructing GE-PI bases using different packages for $l=1, 2, 3, 4$ and for correlation orders $N$'s from $1$ to $8$.
  • Figure 4: Run time for constructing the GE-PI bases using different packages for correlation orders $N = 3, 4, 5, 6$ and for polynomial degrees $l$'s from 1 to 8.
  • Figure 5: Construction time (ms) of the GE-PI bases versus the number of classes in the matrix ${{M}}^{\textup{up}}$\ref{['eq:M2-PI']}, or the number of basis functions.
  • ...and 4 more figures

Theorems & Definitions (80)

  • Definition 2.1: Group-equivariant (GE) function
  • Definition 2.2: Group-equivariant and permutation invariant (GE-PI) function
  • Remark 2.3
  • Proposition 2.4: Basis of the GE space $V^{{\bm l},L}$
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7: Basis of PI functions in $V^{\bm l}$
  • Proposition 2.8: Basis of PI space ${\bar{V}}^{\bm l}$
  • Remark 2.9: Group action
  • ...and 70 more