Variations on the Chebyshev-Lagrange Activation Function

Abstract

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.