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Non Commutative Convolutional Signal Models in Neural Networks: Stability to Small Deformations

arXiv 2310.03879 ↗First posted 5 Oct 2023cs.LGcited by 0

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Abstract (arXiv)

In this paper we discuss the results recently published in~[1] about algebraic signal models (ASMs) based on non commutative algebras and their use in convolutional neural networks. Relying on the general tools from algebraic signal processing (ASP), we study the filtering and stability properties of non commutative convolutional filters. We show how non commutative filters can be stable to small perturbations on the space of operators. We also show that although the spectral components of the Fourier representation in a non commutative signal model are associated to spaces of dimension larger than one, there is a trade-off between stability and selectivity similar to that observed for commutative models. Our results have direct implications for group neural networks, multigraph neural networks and quaternion neural networks, among other non commutative architectures. We conclude by corroborating these results through numerical experiments.

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References · 14

  1. 2019Stability Properties of Graph Neural Networkscited by 127
  2. 2018Quaternion Convolutional Neural Networks for Heterogeneous Image Processingcited by 29
  3. 2020Algebraic Neural Networks: Stability to Deformationscited by 22
  4. 2020Graphon Pooling in Graph Neural Networkscited by 11
  5. 2022Convolutional Learning on Multigraphscited by 10
  6. 2020Discrete Signal Processing with Set Functionscited by 8
  7. 2020Stability of Algebraic Neural Networks to Small Perturbationscited by 8
  8. 2020Discrete Signal Processing on Meet/Join Latticescited by 8
  9. 2021Convolutional Filtering and Neural Networks with Non Commutative Algebrascited by 8
  10. 2022Graphon Pooling for Reducing Dimensionality of Signals and Convolutional Operators on Graphscited by 8
  11. 2022Stability of Aggregation Graph Neural Networkscited by 7
  12. 2023Lie Group Algebra Convolutional Filterscited by 7
2 more
  1. 2022Learning with Multigraph Convolutional Filterscited by 5
  2. 2022Algebraic Convolutional Filters on Lie Group Algebrascited by 4

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