We study the geometrical structure of n by n correlation matrices with rank at most d, with d less than n. We show that they form a stratiefed space. This enables us to develop intrinsic optimization algorithms that efficiently find the nearest low-rank correlation matrix. The algorithm is shown theoretically to be globally convergent to a local minimum, with a quadratic local rate of convergence. This approach allows us to use any norm, in particular the Hadamard norm. This is joint work with Raoul Pietersz from Erasmus University Rotterdam.