Inference algorithms for pattern-based CRFs on sequence data

We consider Conditional Random Fields (CRFs) with pattern-based potentials defined on a chain. In this model the energy of a string (labeling) $x_1 ... x_n$ is the sum of terms over intervals $[i,j]$ where each term is non-zero only if the substring $x_i ... x_j$ equals a prespecified pattern $\alpha$. Such CRFs were used in computer vision, and can be naturally applied to many sequence tagging problems. Let $\Pi$ be the set of input patterns and $L=\sum_{\alpha\in\Pi}|\alpha|$ be their total length. (Komodakis & Paragios, 2009) showed how to compute MAP in time $O(n L)$ when all costs are non-positive. We present a modification that has the same worst-case complexity but can beat it in the best case. More importantly, we give efficient algorithms for the three standard inference tasks in a CRF, namely computing (i) the partition function, (ii) marginals, and (iii) MAP in the general case (i.e. when costs can be positive). Their complexities are respectively $O(n L)$, $O(n\sum_{\alpha\in\Pi}|\alpha|^2)$, and $O(n L\cdot \min{|D|,\log (l_{max}+1)})$ where $D$ is the input alphabet and $l_{max}=\max_{\alpha\in\Pi}|\alpha|$.