The proposal of Daumé III (2009) is that we train a model f aug : {0, 1} (K+1)d → Ω that instead uses an augmented feature vector x aug ∈ {0, 1} (K+1)d defined as where x k ∈ {0, 1} d is a feature representation of x under the k-th domain.
we expand the feature space by K + 1 times by duplicating the original feature x for K times: then given an input from the k-th domain, we only use the original x and the corresponding domain representation x k .