reduce the dependence on $k^2$.

The same issues also arise with a continuous $y$: how to define and estimate the pairs of matrices whose generalized eigenvectors should be extracted is not immediately clear. Still, the case where $y$ is multidimensional (vector regression) can be reduced to the case of univariate $y$ using the same technique of contraction with a vector $u$. Feature extraction from a continuous $y$ can be done by discretization (solely for the purpose of feature extraction), which is much easier in the univariate case than in the multivariate case.

In domains where examples exhibit large variation, or when labeled data is scarce, incorporating prior knowledge is extremely important. For example, in image recognition, convolutions and local pooling are popular ways to generate representations that are invariant to localized distortions. Directly exploiting the spatial or temporal structure of the input signal, as well as incorporating other kinds of invariances in our framework, is a direction for future work.

High dimensional problems create both computational and statistical challenges. Computationally, when $d > 10^6$, the solution of generalized eigenvalue problems can only be performed via specialized libraries such as ScaLAPACK, or via randomized techniques, such as those outlined in [@halko2011finding; 