second moment but different higher order statistics. While our limited experience here suggests second moments are informative for natural datasets, there are potential benefits in using higher order moments. For example, we could replace our class-conditional second moment matrix with a second moment matrix conditioned on other events, informed by higher order moments.

As the number of class labels increases, say $k \geq 1000$, our brute force all-pairs approach, which scales as $O(k^2)$, becomes increasingly difficult both computationally and statistically: we need to solve $O(k^2)$ eigenvector problems (possibly in parallel) and deal with $O(k^2)$ features in the ultimate classifier. Taking a step back, the object of our attention is the tensor ${\mathbb{E}}[x
\otimes x \otimes y]$ and in this paper we only studied one way of selecting pairs of slices from it. In particular, our slices are tensor contractions with one of the standard basis vectors in ${\mathbb{R}}^k$. Clearly, contracting the tensor with any vector $u$ in ${\mathbb{R}}^k$ is possible. This contraction leads to a $d\times d$ second moment matrix which averages the examples of the different classes in the way prescribed by $u$. Any sensible, data-dependent way of picking a good set of vectors $u$ should be able to 