When learning functions on manifolds, we can improve performance by regularizing with respect to the intrinsic manifold geometry rather than the ambient space. However, when regularizing tensor learning, calculating the derivatives along this intrinsic geometry is not possible, and so existing approaches are limited to regularizing in Euclidean space. Our new method for intrinsically regularizing and learning tensors on Riemannian manifolds introduces a surrogate object to encapsulate the geometric characteristic of the tensor. Regularizing this instead allows us to learn non-symmetric and high-order tensors. We apply our approach to the relative attributes problem, and we demonstrate that explicitly regularizing high-order relationships between pairs of data points improves performance.