# Supervised cosine similarity

Suppose we have some samples, each sample is with two vectors and the corresponding label. That is, it looks like ($\mathbf{u}_i, \mathbf{v}_i, y_i$), where $y_i \in \{0, 1\}$ We can calculate the cosine similarity between the two vectors $\mathbf{u}_i$ and $\mathbf{v}_i$. According to the ranking of similarity scores, we can obtain a performance value, such as the Area under the ROC curve (AUC).

However, this may be not the best score and we can do it more better. Since we have the labels, we can train a model to boost the performance of the classifier. I use the following equation and can get a better AUC value.

$\text{score}(\mathbf{u}_i, \mathbf{v}_i)=\frac{\mathbf{u}_i^T\mathbf{v}_i +p}{|\mathbf{u}_i|_2|\mathbf{v}_i|_2}$, where $p$ is learned through training.

So, my question is: what does it mean from $\frac{\mathbf{u}_i\mathbf{v}_i}{|\mathbf{u}_i|_2|\mathbf{v}_i|_2}$ to $\frac{\mathbf{u}_i\mathbf{v}_i+p}{|\mathbf{u}_i|_2|\mathbf{v}_i|_2}$ ? Can we call $\frac{\mathbf{u}_i\mathbf{v}_i+p}{|\mathbf{u}_i|_2|\mathbf{v}_i|_2}$ supervised cosine similarity?