EM-Training for Weighted Aligned Hypergraph Bimorphisms

We develop the concept of weighted aligned hypergraph bimorphism where the weights may, in particular, represent probabilities. Such a bimorphism consists of an R≥0-weighted regular tree grammar, two hypergraph algebras that interpret the generated trees, and a family of alignments between the two interpretations. Semantically, this yields a set of bihypergraphs each consisting of two hypergraphs and an explicit alignment between them; e.g., discontinuous phrase structures and nonprojective dependency structures are bihypergraphs. We present an EM-training algorithm which takes a corpus of bihypergraphs and an aligned hypergraph bimorphism as input and generates a sequence of weight assignments which converges to a local maximum or saddle point of the likelihood function of the corpus.