However, we do not need the word embeddings themselves to verify coplanarity; when there is no reconstruction error, we can express it as a constraint over M, the matrix that is implicitly factorized by the embedding model (see Definition 5).
lemma 2 A linear analogy f : x → x + r holds over a set of ordered pairs S in an SGNS or GloVe word embedding space W with no reconstruction error iff ∃ λ ∈ R, g : x c → x c + λ r holds over S in the corresponding context space C. an analogy f that holds over S in the word space has a corresponding analogy g that holds over S in the context space.