$|v^\top x|^{1/2}$.[^1] In principle, smoothing splines or any other flexible set of univariate basis functions could be used. In our experiments we simply fit a piecewise cubic polynomial on $|v^\top x|^{1/2}$. The polynomial has only two pieces, one for $v^\top x>0$ and one for $v^\top x \leq 0$. We briefly experimented with interaction terms between projection magnitudes, but did not find them beneficial.

Additionally, we need to address from which class pairs to extract eigenvectors. A simple and empirically effective approach, suitable when the number of classes is modest, is to just use all ordered pairs of classes. This can be wasteful if two classes are never confused. The alternative, however, of leaving out a pair $(i,j)$ is that the classifier might have no way of distinguishing between these two classes. Since we do not know upfront which pairs of classes will be confused, our brute force approach is just a safe way to endow the classifier with enough flexibility to deal with any pair of classes that could potentially be confused. Of course, as the number of classes grows, this brute force approach becomes less viable both computationally (due to the quadratic increase in generalized eigenvalue problems) and statistically 