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.

Random Fourier Features (RFF) is among the most popular and broadly applicable approaches for scaling up kernel methods. In essence, RFF allows the user to avoid costly computations with a large kernel matrix via a fast randomized approximation. However, a pervasive difficulty in applying RFF is that the user does not know the actual error of the approximation, or how this error will propagate into downstream learning tasks. Up to now, the RFF literature has primarily dealt with these uncertainties using theoretical error bounds, but from a user’s standpoint, such results are typically impractical—either because they are highly conservative or involve unknown quantities. To tackle these general issues in a data-driven way, this paper develops a bootstrap approach to numerically estimate the errors of RFF approximations. Three key advantages of this approach are: (1) The error estimates are specific to the problem at hand, avoiding the pessimism of worst-case bounds. (2) The approach is flexible with respect to different uses of RFF, and can even estimate errors in downstream learning tasks. (3) The approach enables adaptive computation, in the sense that the user can quickly inspect the error of a rough initial kernel approximation and then predict how much extra work is needed. Furthermore, in exchange for all of these benefits, the error estimates can be obtained at a modest computational cost.

Despite their widespread success, the application of deep neural networks to functional data remains scarce today. The infinite dimensionality of functional data means standard learning algorithms can be applied only after appropriate dimension reduction, typically achieved via basis expansions. Currently, these bases are chosen a priori without the information for the task at hand and thus may not be effective for the designated task. We instead propose to adaptively learn these bases in an end-to-end fashion. We introduce neural networks that employ a new Basis Layer whose hidden units are each basis functions themselves implemented as a micro neural network. Our architecture learns to apply parsimonious dimension reduction to functional inputs that focuses only on information relevant to the target rather than irrelevant variation in the input function. Across numerous classification/regression tasks with functional data, our method empirically outperforms other types of neural networks, and we prove that our approach is statistically consistent with low generalization error.