Neural networks emulate real-valued circuits with explicit complexity bounds controlled by gate count and structure; any definable model with a parallelization condition is a universal approximator precisely when it contains a non-affine nonlinearity.
[47]de Boor, C., and DeVore, R.Approximation by smooth multivariate splines.Transactions of the American Mathematical Society 276, 2 (1983), 775–788
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Algorithmic Foundations of Deep Learning: Complexity-Theoretic Rates and a Characterization of Universal Approximation
Neural networks emulate real-valued circuits with explicit complexity bounds controlled by gate count and structure; any definable model with a parallelization condition is a universal approximator precisely when it contains a non-affine nonlinearity.