  We introduce a novel class of sample-based explanations we term
high-dimensional representers, that can be used to explain the predictions of a
regularized high-dimensional model in terms of importance weights for each of
the training samples. Our workhorse is a novel representer theorem for general
regularized high-dimensional models, which decomposes the model prediction in
terms of contributions from each of the training samples: with positive
(negative) values corresponding to positive (negative) impact training samples
to the model's prediction. We derive consequences for the canonical instances
of $\ell_1$ regularized sparse models, and nuclear norm regularized low-rank
models. As a case study, we further investigate the application of low-rank
models in the context of collaborative filtering, where we instantiate
high-dimensional representers for specific popular classes of models. Finally,
we study the empirical performance of our proposed methods on three real-world
binary classification datasets and two recommender system datasets. We also
showcase the utility of high-dimensional representers in explaining model
recommendations.
