Memory capacity of three-layer neural networks with non-polynomial activations

The minimal number of neurons required for a feedforward neural network to interpolate n generic input-output pairs from Rᵈ× R is Θ(√n). While previous results have shown that Θ(√n) neurons are sufficient, they have been limited to logistic, Heaviside, and rectified linear unit (ReLU) as the activation function. Using a different approach, we prove that Θ(√n) neurons are sufficient as long as the activation function is real analytic at a point and not a polynomial there. Thus, the only practical activation functions that our result does not apply to are piecewise polynomials. Importantly, this means that activation functions can be freely chosen in a problem-dependent manner without loss of interpolation power.