Combining (second-order) graph-based and headed-span-based projective dependency parsing

Graph-based methods, which decompose the score of a dependency tree into scores of dependency arcs, are popular in dependency parsing for decades. Recently, \citet{Yang2022Span} propose a headed-span-based method that decomposes the score of a dependency tree into scores of headed spans. They show improvement over first-order graph-based methods. However, their method does not score dependency arcs at all, and dependency arcs are implicitly induced by their cubic-time algorithm, which is possibly sub-optimal since modeling dependency arcs is intuitively useful. In this work, we aim to combine graph-based and headed-span-based methods, incorporating both arc scores and headed span scores into our model. First, we show a direct way to combine with $O(n^4)$ parsing complexity. To decrease complexity, inspired by the classical head-splitting trick, we show two $O(n^3)$ dynamic programming algorithms to combine first- and second-order graph-based and headed-span-based methods. Our experiments on PTB, CTB, and UD show that combining first-order graph-based and headed-span-based methods is effective. We also confirm the effectiveness of second-order graph-based parsing in the deep learning age, however, we observe marginal or no improvement when combining second-order graph-based and headed-span-based methods. Our code is publicly available at \url{https://github.com/sustcsonglin/span-based-dependency-parsing}.