Fractional optimal control for deep convolutional neural networks exploring ODE-based solutions for image denoising

Abstract

In recent years, Deep Convolutional Neural Networks (DCNNs) have been shown to be effective in low-level vision tasks such as image denoising. DCNN backpropagation is revealed to be characterised as an optimal control problem, where the state equation modelling DCNN forward propagation is an ordinary differential equation. In this article, by studying an optimal fractional control problem, we will develop a neural network whose weights are denoted by $ \theta $ and named $ \theta $-FOCNet which models a generalisation of a DCNN network [21]. So, the aim of this work is to address the control problem $ \theta $-FOCNet for parameters learning in a new nonlinear model with a fractional Caputo derivative of order $ \gamma \in (0, 1) $ in time. To treat this problem, we first analyse the existence and uniqueness of the solution for a nonlinear fractional model (state equation) by using the Schaefer and Banach fixed points theorems, then we prove the existence of an optimal solution for the control problem. In addition, we introduce a non-smooth ADMM (Alternating Direction Method of Multipliers) algorithm for the numerical simulation of the computed free-noise image. Finally, to demonstrate the effectiveness of the proposed method, we present some numerical experiments and compare it with other competitive methods.