Controlled Learning of Pointwise Nonlinearities in Neural-Network-Like Architectures

TL;DR

This paper introduces a variational framework for training nonlinearities in neural architectures with slope constraints, ensuring properties like stability and invertibility, and demonstrates its application in image denoising and inverse problems.

Contribution

It proposes a novel variational approach for learning nonlinearities with slope constraints, using adaptive splines, and applies it to image denoising and inverse problem solving.

Findings

Nonlinearities are optimally represented as adaptive nonuniform linear splines.

The framework enforces properties like 1-Lipschitz stability and invertibility.

Successful application to image denoising and inverse problems.

Abstract

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.