Media theory is a new branch of discrete applied mathematics originally developed in mid-nineties to deal with stochastic evolution of preference relations in political science and mathematical psychology. The theory focuses on a particular semigroup of ‘messages’ acting as transformations of a set of ‘states’, called a ‘medium’, whose axioms are both strong and natural. The term ‘medium’ stems...

In recent years, in view of the transformation in education in the Russian Federation, it has been necessary to introduce different technical facilities and new educational methods into the educational process. An introduction of training systems into the educational process concerns every educational method. New methods use various results from such fields as pedagogy, psychology, cybernetics,...

We study the convergence properties of a pair of learning algorithms (learning with and without memory). This leads us to study the dominant eigenvalue of a class of random matrices. This turns out to be related to the roots of the derivative of random polynomials (generated by picking their roots uniformly at random in the interval [0, 1], although our results extend to other distributions). T...

Recently there has been a dramatic increase in the performance of recognition systems due to the introduction of deep architectures for representation learning and classification. However, the mathematical reasons for this success remain elusive. This tutorial will review recent work that aims to provide a mathematical justification for several properties of deep networks, such as global optima...