Inference Algorithms for Pattern-Based CRFs on Sequence Data

Abstract

We consider Conditional random fields (CRFs) with pattern-based potentials defined on a chain. In this model the energy of a string (labeling) $x_1\ldots x_n$ is the sum of terms over intervals [i, j] where each term is non-zero only if the substring $x_i\ldots x_j$ equals a prespecified pattern w. Such CRFs can be naturally applied to many sequence tagging problems. We present efficient algorithms for the three standard inference tasks in a CRF, namely computing (i) the partition function, (ii) marginals, and (iii) computing the MAP. Their complexities are respectively $O(\textit{nL})$, $O(\textit{nL} \ell _{\max })$ and $O(\textit{nL} \min \{|D|,\log (\ell _{\max }\!+\!1)\})$ where L is the combined length of input patterns, $\ell _{\max }$ is the maximum length of a pattern, and D is the input alphabet. This improves on the previous algorithms of Ye et al. (NIPS, 2009) whose complexities are respectively $O(\textit{nL} |D|)$, $O\left( n |\varGamma | L^2 \ell _{\max }^2\right)$ and $O(\textit{nL} |D|)$, where $|\varGamma |$ is the number of input patterns. In addition, we give an efficient algorithm for sampling, and revisit the case of MAP with non-positive weights.