  We seek to improve the data efficiency of neural networks and present novel
implementations of parameterized piece-wise polynomial activation functions.
The parameters are the y-coordinates of n+1 Chebyshev nodes per hidden unit and
Lagrangian interpolation between the nodes produces the polynomial on [-1, 1].
We show results for different methods of handling inputs outside [-1, 1] on
synthetic datasets, finding significant improvements in capacity of expression
and accuracy of interpolation in models that compute some form of linear
extrapolation from either ends. We demonstrate competitive or state-of-the-art
performance on the classification of images (MNIST and CIFAR-10) and
minimally-correlated vectors (DementiaBank) when we replace ReLU or tanh with
linearly extrapolated Chebyshev-Lagrange activations in deep residual
architectures.
