We present a unifying functional framework for the implementation and training of deep neural networks (DNN) with free-form activation functions. To make the problem well posed, we constrain the shape of the trainable activations (neurons) by penalizing their second-order total-variations. We prove that the optimal activations are adaptive piecewise-linear splines, which allows us to recast the problem as a parametric optimization.
We then specify some corresponding trainable B-spline-based activation units. These modules can be inserted in deep neural architectures and optimized efficiently using standard tools. We provide experimental results that demonstrate the benefit of our approach.
This is joint work with Pakshal Bohra, Joaquim Campos, Harshit Gupta, Shayan Aziznejad.