Low-Rank Approximation of Weighted Tree Automata
This provides a more stable and effective minimization technique for formalisms like probabilistic context-free grammars, which are used in natural language processing.
The paper tackles the problem of minimizing weighted tree automata (WTA) by developing an efficient algorithm based on singular value decomposition of infinite Hankel matrices, achieving lower perplexity than previous methods on real-world treebank data.
We describe a technique to minimize weighted tree automata (WTA), a powerful formalisms that subsumes probabilistic context-free grammars (PCFGs) and latent-variable PCFGs. Our method relies on a singular value decomposition of the underlying Hankel matrix defined by the WTA. Our main theoretical result is an efficient algorithm for computing the SVD of an infinite Hankel matrix implicitly represented as a WTA. We provide an analysis of the approximation error induced by the minimization, and we evaluate our method on real-world data originating in newswire treebank. We show that the model achieves lower perplexity than previous methods for PCFG minimization, and also is much more stable due to the absence of local optima.