(due to the increase in the number of features for the final classifier). We discuss issues regarding large numbers of classes in Section \[sec:discussion\].

Finally, the generalized eigenvalues can guide us in picking a subset of the $d$ generalized eigenvectors we could extract from each class pair, i.e., generalized eigenvalues are useful for feature selection. A generalized eigenvector $v$ with eigenvalue $\lambda$ has ${\mathbb{E}}[(v^\top x)^2 | y]$ equal to $1$ for the denominator class $y=j$ and equal to $\lambda$ for the numerator class $y=i$. Therefore, eigenvalues far from 1 correspond to highly discriminative features. Similar to [@platt2010translingual], we extract the top few eigenvectors, as top eigenspaces are cheaper to compute than bottom eigenspaces. To guard against picking non-discriminative eigenvectors, we discard those whose eigenvalues are less than a threshold $\theta>1$.

The above observations lead to the GEM procedure outlined in Algorithm \[alg:gem\]. Although Algorithm \[alg:gem\] has proven sufficiently versatile for the experiments described herein, it is merely an example of how to use generalized eigenvalue based features for multiclass classification. Other classification techniques could benefit from using the raw projection values without any nonlinear manipulation, e.g., decision trees; additionally the generalized eigenvectors could be used to initialize a neural network 