A local feature tensor similarity based deep learning approach is proposed in this paper for 3D face recognition. Once a set of salient points on the 3D mesh are detected, three scale and rotation invariant features are extracted to represent local surface around each salient point. The local features of all the salient points are concatenated to produce a 3rd order feature tensor to represent a 3D face. Similarity of two 3D faces can thus be measured by a similarity tensor calculated using the two feature tensors. To address the unavailability of large 3D face samples, a feature tensor based data augmentation approach is proposed to augment the number of feature tensors. Experimental results show that the ResNet model trained using the augmented feature tensors achieves the best performance among state of the art competitors, i.e. 99.71% and 96.2% accuracy are achieved for Bosphorus and BU3DFE database, respectively.