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The Geometry of Polynomial Group Convolutional Neural Networks

Yacoub Hendi, Daniel Persson, Magdalena Larfors

Abstract

We study polynomial group convolutional neural networks (PGCNNs) for an arbitrary finite group $G$. In particular, we introduce a new mathematical framework for PGCNNs using the language of graded group algebras. This framework yields two natural parametrizations of the architecture, based on Hadamard and Kronecker products, related by a linear map. We compute the dimension of the associated neuromanifold, verifying that it depends only on the number of layers and the size of the group. We also describe the general fiber of the Kronecker parametrization up to the regular group action and rescaling, and conjecture the analogous description for the Hadamard parametrization. Our conjecture is supported by explicit computations for small groups and shallow networks.

The Geometry of Polynomial Group Convolutional Neural Networks

Abstract

We study polynomial group convolutional neural networks (PGCNNs) for an arbitrary finite group . In particular, we introduce a new mathematical framework for PGCNNs using the language of graded group algebras. This framework yields two natural parametrizations of the architecture, based on Hadamard and Kronecker products, related by a linear map. We compute the dimension of the associated neuromanifold, verifying that it depends only on the number of layers and the size of the group. We also describe the general fiber of the Kronecker parametrization up to the regular group action and rescaling, and conjecture the analogous description for the Hadamard parametrization. Our conjecture is supported by explicit computations for small groups and shallow networks.

Paper Structure

This paper contains 20 sections, 18 theorems, 107 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. Related works.
  4. Notation.
  5. Preliminaries on graded group algebras
  6. Graded group algebras and regular representations
  7. Kronecker and Hadamard products
  8. General conditions and symbolic ranks
  9. Invertibility of filters
  10. Polynomial group convolutional neural networks
  11. Definition of polynomial group convolutional neural networks
  12. Neuromanifolds of PGCNNs
  13. Main results on neuromanifolds of PGCNN
  14. Dimension of $\mathcal{M}_{\varphi}$ and $\mathcal{M}_{\Phi}$
  15. General fiber of $\varphi$
  16. ...and 5 more sections

Key Result

Proposition 2.7

Let $\theta \in S[G]$. If $\text{Mat}_\theta$ is invertible, then its inverse is also $G$-circulent. By Remark remark: Mat_is_rep, $\text{Mat}_\theta$ is invertible if and only if $\theta$ is invertible in $S[G]$.

Figures (2)

  • Figure 1: PGCNN parametrization maps $\varphi$ and $\Phi$.
  • Figure 2: A commutative diagram that describes the relation between the projectivized parametrization maps $\tilde{\Phi}$ and $\tilde{\varphi}$ induced by the PGCNN architecture with $L$ layers and activation degree $r$ over the finite group $G$.

Theorems & Definitions (58)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4: $S\lbrack G\rbrack$ regular representation
  • Definition 2.5: $G$-circulent matrix
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8: Definition of Kronecker Product
  • Remark 2.9
  • ...and 48 more