V = (\hat C_j+\frac \gamma d \operatorname*{Trace}(\hat C_j) I) V \Lambda $ $F \leftarrow F \cup \{ V_{q} | \Lambda_{qq} \geq \theta \}$ $\psi_{v,\alpha,\delta}(x) \doteq \max(0,\delta v^\top x)^{\alpha/2}$\
$\phi(x) \doteq [\psi_{v,\alpha,\delta}(x) | v,\alpha,\delta \in F \times \{1,2,3\} \times \{-1,1\}]$\
$w = \mathrm{MultiLogit} (\{ (\phi(x), y) | (x, y) \in S \})$

We are left with specifying a full algorithm for multiclass classification. First we need to specify how to use the eigenvectors $\{ v_i \}$. The eigenvectors define an embedding for each example $x$ using the projection magnitudes $\{ v_i^\top x \}$ as new coordinates. However the embedding is linear, therefore composition with a linear classifier is equivalent to learning a linear classifier in the original space, perhaps with a different regularization. This motivates the use of nonlinear functions of the projection magnitude.

To construct nonlinear maps, we can get inspiration from the optimization criterion in equation , i.e., the ratio of expected projection magnitudes conditional on different class labels. For example, we could use a nonlinear map such as $(v^\top x)^2$. This type of nonlinearity can be sensitive (for example, it is not Lipschitz) so in practice more robust proxies can be used such as $|v^\top x|$ or even 