Field Embedded Factorization Machine

Field Embedded Factorization Machine, or FEFM, is a factorization machine variant. For each field pair, FEFM introduces symmetric matrix embeddings along with the usual feature vector embeddings that are present in FM. Like FM, $v_{i}$ is the vector embedding of the $i^{t h}$ feature. However, unlike Field-Aware Factorization Machines (FFMs), FEFM doesn't explicitly learn field-specific feature embeddings. The learnable symmetric matrix $W_{F(i), F(j)}$ is the embedding for the field pair $F(i)$ and $F(j) .$ The interaction between the $i^{t h}$ feature and the $j^{t h}$ feature is mediated through $W_{F(i), F(j)} .$

$$ \phi(\theta, x)=\phi_{F E F M}((w, v, W), x)=w_{0}+\sum_{i=1}^{m} w_{i} x_{i}+\sum_{i=1}^{m} \sum_{j=i+1}^{m} v_{i}^{T} W_{F(i), F(j)} v_{j} x_{i} x_{j} $$

where $W_{F(i), F(j)}$ is a $k \times k$ symmetric matrix ( $k$ is the dimension of the feature vector embedding space containing feature vectors $v_{i}$ and $v_{j}$ ).

The symmetric property of the learnable matrix $W_{F(i), F(j)}$ is ensured by reparameterizing $W_{F(i), F(j)}$ as $U_{F(i), F(j)}+$ $U_{F(i), F(j)}^{T}$, where $U_{F(i), F(j)}^{T}$ is the transpose of the learnable matrix $U_{F(i), F(j)} .$ Note that $W_{F(i), F(j)}$ can also be interpreted as a vector transformation matrix which transforms a feature embedding when interacting with a specific field.