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, vᵢ 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)) .

ϕ(θ, x)=ϕ_(F E F M)((w, v, W), x)=w₀+∑ᵢ₌₁ᵐ wᵢ xᵢ+∑ᵢ₌₁ᵐ ∑ⱼ₌ᵢ₊₁ᵐ vᵢᵀ W_(F(i), F(j)) vⱼ xᵢ xⱼ

where W_(F(i), F(j)) is a k ×k symmetric matrix ( k is the dimension of the feature vector embedding space containing feature vectors vᵢ and vⱼ ).

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.