The paper proposes a practical method for a significant dimensionality reduction of Volterra kernels, defining a discrete nonlinear model of a signal by Volterra series of higher order. In system identification of Volterra series, the Volterra kernels and nonlinear inputs of the system can be described by super-symmetrical tensors. The reduction of their dimensionality is obtained by a tensor decomposition technique called Higher Order Singular Value Decomposition (HOSVD). The main contribution of the paper is a cascade learning algorithm for the system identification based on residuals of least squares minimization. Numerical examples for Volterra system of order four are used to illustrate the approach.