SAND: Smooth imputation of sparse and noisy functional data with Transformer networks

Although the transformer architecture has come to dominate other models for text and image data, its application to irregularly-spaced longitudinal data has been limited. We introduce a variant of the transformer that enables it to more smoothly impute such functional data. We augment the vanilla transformer with a simple module we call SAND (self-attention on derivatives), which naturally encourages smoothness by modeling the sub-derivative of the imputed curve. On the theoretical front, we prove the number of hidden nodes required by a network with SAND to achieve an $\epsilon$ prediction error bound for functional imputation. Extensive experiments over various types of functional data demonstrate that transformers with SAND produce better imputations than both their standard counterparts as well as transformers augmented with alternative approaches to encode the inductive bias of smoothness. SAND also outperforms standard statistical methods for functional imputation like kernel smoothing and PACE.