Analogy preservation is necessary and sufficient for a linear cross-lingual map

Peng et al. prove a necessary-and-sufficient equivalence between within-language analogy (parallelogram) preservation and linearity of the ground-truth cross-lingual embedding map [peng-etal-2020-understanding-linearity-cross-lingual-mappings] They test it against real monolingual embeddings across 12 languages (fastText-Wiki, fastText-CommonCrawl, word2vec-CoNLL) using a new 5-category cross-lingual analogy dataset [peng-etal-2020-understanding-linearity-cross-lingual-mappings] Mapping-linearity quality tracks analogy-preservation quality empirically [peng-etal-2020-understanding-linearity-cross-lingual-mappings] No causal intervention is performed [peng-etal-2020-understanding-linearity-cross-lingual-mappings]