A simplex shrink-wrap algorithm

An iterative algorithm for fitting a simplex around a set of data vectors is proposed. The algorithm is a gradient descent on an objective function defined on the vertices. This objective function is the sum of two terms: the first term is the volume of the simplex, which we would like to minimize, and the second term is a penalty term which has the effect of "pushing" the faces of the simplex away fr om the data points. The gradient of each of these terms is determined analytically, and used in the gradient descent algorithm. The penalty term includes a multiplicative constant which approaches zero as the gradient descent algorithm progresses, causing the iterates to converge to the vertices of a simplex which fits tightly around the given data points. The performance of this algorithm is demonstrated using simulated data generated from real spectral libraries.