# second order regularisation on a neurofuzzy network with Bernstein basis functions

We're trying to build a neural network that uses a neurofuzzy approach. Our reference is the book Adaptive Modelling, Estimation and Fusion from Data: A Neurofuzzy Approach by Chris Harris, Xia Hong, and Qiang Gan.

We're trying to compute the second order regularisation, which becomes a problem of finding a matrix K, which is related to the prior, is maybe called the smoother matrix, and "represents the square of the sum of the curvature evaluated at the centre of the basis functions". The Hessian is equal to

    H = beta*A(transpose)*A + alpha*K


where A is a matrix from the basis functions and K is the matrix we're having trouble with. alpha/beta is the regularisation coefficient.

All the resources I've come across suggest that K can be computed from the centres of the basis functions, except our basis functions are Bernstein polynomials, which are not radial basis functions.

Does anyone have any insight into how to compute K? Does anyone know of any examples of computing K when the basis functions aren't radial?

Thank you so much