Material for : Spectral Unsupervised Parsing with Additive Tree Metrics

] The primary purpose of the supplemental is to provide the theoretical arguments that our algorithm is correct. We first give the proof that our proposed tree metric is indeed tree additive. We then analyze the consistency of Algorithm 1. 1 Path Additivity We first prove that our proposed tree metric is path additive based on the proof technique in Song et al. (2011). Lemma 1. If Assumption 1 in the main paper holds then, d spectral is an additive metric. Proof. For conciseness, we simply prove the property for paths of length 2. The proof for more general cases follows similarly (e.g. see Anandkumar et al. (2011)). First note that the relationship between eigenvalues and singular values allows us to rewrite the distance metric as d spectral (i, j) = − 1 2 log Λ m (Σ x (i, j)Σ x (i, j)) +