A dynamical system perspective of structural learning with forgetting

Structural learning with forgetting is an established method of using Laplace regularization to generate skeletal artificial neural networks. In this paper we develop a continuous dynamical system model of regularization in which the associated regularization parameter is generalized to be a time-varying function. Analytic results are obtained for a Laplace regularizer and a quadratic error surface by solving a different linear system in each region of the weight space. This model also enables a comparison of Laplace and Gaussian regularization. Both of these regularizers have a greater effect in weight space directions which are less important for minimization of a quadratic error function. However, for the Gaussian regularizer, the regularization parameter modifies the associated linear system eigenvalues, in contrast to its function as a control input in the Laplace case. This difference provides additional evidence for the superiority of the Laplace over the Gaussian regularizer.