Controlled Learning of Pointwise Nonlinearities in Neural-Network-Like Architectures
This work addresses the need for stable and invertible nonlinearities in signal-processing algorithms, though it appears incremental as it builds on existing variational frameworks with added constraints.
The authors tackled the problem of training freeform nonlinearities in neural networks with slope constraints to ensure stability and other properties, achieving global optima with adaptive nonuniform linear splines and demonstrating applications in image denoising and inverse problems.
We present a general variational framework for the training of freeform nonlinearities in layered computational architectures subject to some slope constraints. The regularization that we add to the traditional training loss penalizes the second-order total variation of each trainable activation. The slope constraints allow us to impose properties such as 1-Lipschitz stability, firm non-expansiveness, and monotonicity/invertibility. These properties are crucial to ensure the proper functioning of certain classes of signal-processing algorithms (e.g., plug-and-play schemes, unrolled proximal gradient, invertible flows). We prove that the global optimum of the stated constrained-optimization problem is achieved with nonlinearities that are adaptive nonuniform linear splines. We then show how to solve the resulting function-optimization problem numerically by representing the nonlinearities in a suitable (nonuniform) B-spline basis. Finally, we illustrate the use of our framework with the data-driven design of (weakly) convex regularizers for the denoising of images and the resolution of inverse problems.