This paper introduces a new framework for surface analysis derived from the general setting of elastic Riemannian metrics on shape spaces. Traditionally, those metrics are defined over the infinite dimensional manifold of immersed surfaces and satisfy specific invariance properties enabling the comparison of surfaces modulo shape preserving transformations such as reparametrizations. The specificity of our approach is to restrict the space of allowable transformations to predefined finite dimensional bases of deformation fields. These are estimated in a data-driven way so as to emulate specific types of surface transformations. This allows us to simplify the representation of the corresponding shape space to a finite dimensional latent space. However, in sharp contrast with methods involving e.g. mesh autoencoders, the latent space is equipped with a non-Euclidean Riemannian metric inherited from the family of elastic metrics. We demonstrate how this model can be effectively implemented to perform a variety of tasks on surface meshes which, importantly, does not assume these to be pre-registered or to even have a consistent mesh structure. We specifically validate our approach on human body shape and pose data as well as human face and hand scans for problems such as shape registration, interpolation, motion transfer or random pose generation.