  Stratified models depend in an arbitrary way on a selected categorical
feature that takes $K$ values, and depend linearly on the other $n$ features.
Laplacian regularization with respect to a graph on the feature values can
greatly improve the performance of a stratified model, especially in the
low-data regime. A significant issue with Laplacian-regularized stratified
models is that the model is $K$ times the size of the base model, which can be
quite large.
  We address this issue by formulating eigen-stratifed models, which are
stratified models with an additional constraint that the model parameters are
linear combinations of some modest number $m$ of bottom eigenvectors of the
graph Laplacian, i.e., those associated with the $m$ smallest eigenvalues. With
eigen-stratified models, we only need to store the $m$ bottom eigenvectors and
the corresponding coefficients as the stratified model parameters. This leads
to a reduction, sometimes large, of model size when $m \leq n$ and $m \ll K$.
In some cases, the additional regularization implicit in eigen-stratified
models can improve out-of-sample performance over standard Laplacian
regularized stratified models.
