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Universal approximations of permutation invariant/equivariant functions by deep neural networks

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arxiv 1903.01939 v3 pith:OSQI55QW submitted 2019-03-05 cs.LG stat.ML

classification cs.LGstat.ML
keywords equivariantinvariantdeepneuralapproximatorfunctionsnetworkstheory
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abstract

In this paper, we develop a theory about the relationship between $G$-invariant/equivariant functions and deep neural networks for finite group $G$. Especially, for a given $G$-invariant/equivariant function, we construct its universal approximator by deep neural network whose layers equip $G$-actions and each affine transformations are $G$-equivariant/invariant. Due to representation theory, we can show that this approximator has exponentially fewer free parameters than usual models.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions

    cs.LG 2025-11 conditional novelty 7.0 of 10

    Symmetric squared-ReLU networks approximate symmetric Korobov functions at rate O(m^{-1}) with a dimension-independent prefactor, and gradient-based learning achieves M^{-2/3} generalization error.

  2. When Attention is Beneficial for Learning Wireless Resource Allocation Efficiently?

    eess.SP 2025-07 conditional novelty 6.0 of 10

    Attention appears in a learned wireless policy only along the dimension with interference when that interference is not already present in the environmental parameters.

  3. Transformers Are Universally Consistent

    cs.LG 2025-05 reject novelty 4.0 of 10

    HyT, a hyperbolic Transformer, is claimed to be universally consistent for L2 regression, but the proof is invalidated by an algebraic error and circular reasoning.

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