Estimating intrinsic dimension by sparse convex representation

Abstract

In this paper, a novel sparse convex representation learning algorithm is proposed for estimating the intrinsic dimension of a dataset. Caratheodory's theorem states that if a point x of Rd lies in the convex hull of a set P, there is a subset of P consisting of d + 1 or fewer points such that x lies in the convex hull of P'. We believe that the maximum value, among the numbers of the nonzero elements of the sparsest convex representation of all points, implies the intrinsic dimension of a data set. The sparsest convex representation of a point lying in a convex hull means that it is a convex combination of the minimum number of the extreme points. Based on this basic idea, we constructed an objective function. Moreover, an improved orthogonal matching pursuit (OMP) method is proposed for solving it to derive a sparse convex representation. The obtained solutions can be used for estimating the dimension of the data set. The experiment results show the effectiveness and efficiency of our proposed method.