Word-analogy linearity follows from a paraphrase definition

- Defining a "word transformation" as a paraphrase relation (KL-divergence of induced context distributions) and an analogy as two pairs sharing that transformation, the relation king-man+woman ≈ queen follows as a theorem, up to bounded paraphrase and dependence error terms. [allen-hospedales-2019-analogies-explained] - The account is a mathematical derivation over SGNS embeddings understood as a shifted-PMI factorization. [allen-hospedales-2019-analogies-explained] - Empirical support is a single illustrative example (the man:king::woman:queen residual decomposed into conditional-independence, independence and paraphrase errors on the text8 corpus, Figure 5); no analogy-benchmark accuracy is reported. [allen-hospedales-2019-analogies-explained] - Embeddings are the authors' own word2vec/SGNS trained on text8; an independent follow-up (Fournier et al. 2021) disputes the paraphrase explanation, so the finding is single-source. [allen-hospedales-2019-analogies-explained]