  Influence functions estimate effect of individual data points on predictions
of the model on test data and were adapted to deep learning in Koh and Liang
[2017]. They have been used for detecting data poisoning, detecting helpful and
harmful examples, influence of groups of datapoints, etc. Recently, Ilyas et
al. [2022] introduced a linear regression method they termed datamodels to
predict the effect of training points on outputs on test data. The current
paper seeks to provide a better theoretical understanding of such interesting
empirical phenomena. The primary tool is harmonic analysis and the idea of
noise stability. Contributions include: (a) Exact characterization of the
learnt datamodel in terms of Fourier coefficients. (b) An efficient method to
estimate the residual error and quality of the optimum linear datamodel without
having to train the datamodel. (c) New insights into when influences of groups
of datapoints may or may not add up linearly.
