In order to measure the impact of the embedding dimension on the models performance different values were tried, d = $d_{opt}$, $0.5d_{opt}$, 2$d_{opt}$. For each pair from these three pairwise skill distance was calculated as:

$\xi_{d_1, d_2} = \frac {\sum_{i \gt j}|pdist_{d1} (s_i, s_j) - pdist_{d_2}(s_i, s_j)|} {\begin{pmatrix} N\\ 2 \end {pmatrix}}$ $d_1 \neq d_2$

$pdist_d(s_i, s_j)$ refers to the pairwise distance and N is number of skills. Then the $\xi$ is compared to average pairwie distance $\mu$:

$\mu_d = \frac {\sum_{i \gt j}pdist_{d} (s_i, s_j) } {\begin{pmatrix} N\\ 2 \end {pmatrix}}$ According to this experiment, KQN learned the skill relationship in a better manner when d was higher. Additionally, the authors used Mantel tests to measure the similarity between 2 distance matrices. It showed that when predicting the probability of the correctness as well as for learning relation between skill vectors KQN model is stable for different d values.