  Least-squares models such as linear regression and Linear Discriminant
Analysis (LDA) are amongst the most popular statistical learning techniques.
However, since their computation time increases cubically with the number of
features, they are inefficient in high-dimensional neuroimaging datasets.
Fortunately, for k-fold cross-validation, an analytical approach has been
developed that yields the exact cross-validated predictions in least-squares
models without explicitly training the model. Its computation time grows with
the number of test samples. Here, this approach is systematically investigated
in the context of cross-validation and permutation testing. LDA is used
exemplarily but results hold for all other least-squares methods. Furthermore,
a non-trivial extension to multi-class LDA is formally derived. The analytical
approach is evaluated using complexity calculations, simulations, and
permutation testing of an EEG/MEG dataset. Depending on the ratio between
features and samples, the analytical approach is up to 10,000x faster than the
standard approach (retraining the model on each training set). This allows for
a fast cross-validation of least-squares models and multi-class LDA in
high-dimensional data, with obvious applications in multi-dimensional datasets,
Representational Similarity Analysis, and permutation testing.
