= {\mathbb{E}}[(v_1^\top x)(v_2^\top x)|y=j].  $

Diversity indicates the different generalized eigenvectors per class pair provide complementary information, and that techniques which only use the first generalized eigenvector are not maximally exploiting the data.

Finite Sample Considerations
----------------------------

Even though we have shown the properties of our method assuming knowledge of the expectations ${\mathbb{E}}[xx^\top|y=m]$, in practice we estimate these quantities from our training samples. The empirical average $$\label{eq:empirical}
\hat C_m = \frac{\sum_{i=1}^n {\mathbb{I}}[y_i=m] x_i x_i^\top}{\sum_{i=1}^n {\mathbb{I}}[y_i=m]}$$ converges to the expectation at a rate of $O(n^{-1/2})$. Here and below we are suppressing the dependence upon the dimensionality $d$, which we consider fixed. Typical finite sample tail bounds become meaningful once $n=O(d\log d)$ [@vershynin2010introduction].

Given $\hat C_m= C_m +E_m$ with $||E_m||_2=O(n^{-1/2})$, we can use results from matrix perturbation theory to establish that our finite sample results cannot be too far from those obtained using the expected values. For example, if the *Crawford number* $$c(C_i,C_j) \doteq \min_{||v||=1} (v^\top C_i v)^2 + (v^\top C_j v)^2 > 0,$$ and the perturbations $E_i$ and $E_j$ satisfy $$||E_i||_2^2+||E_j||_2^2 < c(C_i,C_j),$$ then [@golub2012matrix] for all $q \in [d]$ $$\tan(|\tan^{-1}(\lambda_q) - \tan^{-1}(\hat \lambda_q)|) \leq
 O\left(\frac{1}{\sqrt n c(C_i,C_j)}\right),$$ where $\lambda_q,\hat \lambda_q$ are the $q$-th generalized eigenvalues of the matrix pairs $C_i,C_j$ 