The 3D Morphable Model (3DMM) is a powerful statistical tool for representing 3D face shapes. To build a 3DMM, a training set of scans in full point-to-point correspondence is required, and its modeling capabilities directly depend on the variability of the training data. Hence, to increase the descriptive power of a 3DMM, accurately establishing dense correspondence across heterogeneous scans with sufficient diversity in terms of identities, ethnicities, or expressions becomes essential. In this manuscript, we present a fully automatic approach that leverages a 3DMM to establish a dense correspondence across raw 3D faces. We propose a novel formulation to learn a set of sparse deformation components with local support on the face that, together with an original non-rigid deformation algorithm, allow the 3DMM to precisely fit unseen faces and transfer its semantic annotation to arbitrary 3D faces. We experimented our approach on three large and diverse datasets, showing it can effectively generalize to very different samples and accurately establish a dense correspondence even in presence of complex facial expressions. The accuracy of the dense registration is demonstrated by building a heterogeneous, large-scale 3DMM from more than 9,000 fully registered scans obtained by joining the three datasets.