Achieving a greater understanding of natural language syntax and parsing is a critical step in producing useful natural language processing systems. In this thesis, we focus on the formalism of dependency grammar as it allows one to model important head-modifier relationships with a minimum of extraneous structure. Recent research in dependency parsing has highlighted the discriminative structured prediction framework (McDonald et al., 2005a; Carreras, 2007; Suzuki et al., 2009), which is characterized by two advantages: first, the availability of powerful discriminative learning algorithms like log-linear and max-margin models (Lafferty et al., 2001; Taskar et al., 2003), and second, the ability to use arbitrarily-defined feature representations. This thesis explores three advances in the field of discriminative dependency parsing. First, we show that the classic Matrix-Tree Theorem (Kirchhoff, 1847; Tutte, 1984) can be applied to the problem of non-projective dependency parsing, enabling both log-linear and max-margin parameter estimation in this setting. Second, we present novel third-order dependency parsing algorithms that extend the amount o
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