Convergence of Discrete-Time Cellular Neural Networks with Application to Image Processing

The paper considers a class of discrete-time cellular neural networks (DT-CNNs) obtained by applying Euler's discretization scheme to standard CNNs. Let T be the DT-CNN interconnection matrix which is defined by the feedback cloning template. The paper shows that a DT-CNN is convergent, i.e. each solution tends to an equilibrium point, when T is symmetric and, in the case where T + En is not positive-semidefinite, the step size of Euler's discretization scheme does not exceed a given bound (En is the n × n unit matrix). It is shown that two relevant properties hold as a consequence of the local and space-invariant interconnecting structure of a DT-CNN, namely: (1) the bound on the step size can be easily estimated via the elements of the DT-CNN feedback cloning template only; (2) the bound is independent of the DT-CNN dimension. These two properties make DT-CNNs very effective in view of computer simulations and for the practical applications to high-dimensional processing tasks. The obtained results are proved via Lyapunov approach and LaSalle's Invariance Principle in combination with some fundamental inequalities enjoyed by the projection operator on a convex set. The results are compared with previous ones in the literature on the convergence of DT-CNNs and also with those obtained for different neural network models as the Brain-State-in-a-Box model. Finally, the results on convergence are illustrated via the application to some relevant 2D and 1D DT-CNNs for image processing tasks

The Shallow and the Deep:A biased introduction to neural networks and old school machine learning

The Shallow and the Deep is a collection of lecture notes that offers an accessible introduction to neural networks and machine learning in general. However, it was clear from the beginning that these notes would not be able to cover this rapidly changing and growing field in its entirety. The focus lies on classical machine learning techniques, with a bias towards classification and regression. Other learning paradigms and many recent developments in, for instance, Deep Learning are not addressed or only briefly touched upon.Biehl argues that having a solid knowledge of the foundations of the field is essential, especially for anyone who wants to explore the world of machine learning with an ambition that goes beyond the application of some software package to some data set. Therefore, The Shallow and the Deep places emphasis on fundamental concepts and theoretical background. This also involves delving into the history and pre-history of neural networks, where the foundations for most of the recent developments were laid. These notes aim to demystify machine learning and neural networks without losing the appreciation for their impressive power and versatility