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$l_1 \in \mathbb R^{D_1}$ is the output of the inner product layer, where $D_1$ is the dimension of the layer. The formulation of $l_1$ is $$l_1 = relu(l_z + l_p + b_1)$$
herein, with $l_z$ the linear signals, $l_p$ the quadratic signals and $b_1$ the bias.
The linear signals can be obtained by below. $$l_z^n = W_z^n\odot z = \displaystyle\sum_{i=1}^N \displaystyle\sum_{j=1}^M W_{z_{i, j}}^n z_{i,j}$$
Quadratic signals can be obtained by below. $$l_p^n = W_p^n\odot p = \displaystyle\sum_{i-1}^N\displaystyle\sum_{j=1}^M\theta_i^n \theta_j^n \langle f_i, f_j\rangle = \langle \displaystyle\sum_{i=1}^N\delta_i^n,\displaystyle\sum_{i=1}^N\delta_i^n \rangle$$
Evaluation
Evaluation is omitted as the paper approves that the AUC performs better than those without product layer.
input data
Train and target data are provided in the data folder.
They are preprocessed data in percentage of display percentage and CTR for a certain category.