Let G = G(V, E) be an undirected graph where V is a set of vertices and E a set of edges. Let m = (mi, win) be the set of mentions of a document with n mentions to resolve. Each mention rrii in the document is represented as a vertex Vi G V in the graph. An edge e E is added to the graph for pairs of vertices (Vi, Vj) representing the possibility that both mentions corefer.

Let C be our set of constraints. Given a pair of mentions (rrii, rrij), a subset of constraints CV, ç C restrict the compatibility of both mentions. Cij is used to compute the weight value of the edge connecting Vi and Vj. Let Wij G If be the weight of the edge :

where /&(•) is a function that evaluates the con­straint k and is the weight associated to the constraint. Note that and Wij can be negative.

In our approach, each vertex (Vi) in the graph is a variable (Vi) for the algorithm. Let Li be the number of different values (labels) that are possi­ble for v%. The possible labels of each variable are the partitions that the vertex can be assigned. A vertex with index i can be in the first i partitions (i.e. Li = i).

Each pair of mentions (rrii, rrij) in a training doc­ument is evaluated by the set of feature functions shown in Figure 1. The values returned by these functions form a positive example when the pair of mentions corefer, and a negative one otherwise. Three specialized models are constructed depend­ing on the type of anaphor mention (rrij) of the pair: pronoun, named entity or nominal.

A decision tree is generated for each specialized model and a set of rules is extracted with C4.5 rule-learning algorithm (Quinlan, 1993). These rules are our set of constraints. The C4.5rules al­gorithm generates a set of rules for each path from the learned tree. It then checks if the rules can be generalized by dropping conditions.

Given the training corpus, the weight of a con­straint Ck is related with the number of exam­ples where the constraint applies Ack and how many of them corefer Cck- We define A& as

A.

Relaxation labeling (Relax) is a generic name for a family of iterative algorithms which perform function optimization, based on local informa­tion (Hummel and Zucker, 1987). The algorithm solves our weighted constraint satisfaction prob­lem dealing with the edge weights. In this manner, each vertex is assigned to a partition satisfying as many constraints as possible. To do that, the al­gorithm assigns a probability for each possible la­bel of each variable. Let H = (h, h,..., hra) be the weighted labeling to optimize, where each h* is a vector containing the probability distribution of Vi, that is: If = (h\,h%2,..., hlL.). Given that the resolution process is iterative, the probability for label I of variable Vi at time step t is h\(t), or simply h\ when the time step is not relevant.

Initialize:

Main loop: repeat For each variable v..

For each possible label I for v.,

EiEi^j wa x K

End for End for Until no more significant changes

Figure 2: Relaxation labeling algorithm

The support for a pair variable-label (Su) ex­presses how compatible is the assignment of la­bel I to variable Vi taking into account the labels of adjacent variables and the edge weights. The support is defined as the sum of the edge weights that relate variable Vi with each adjacent variable Vj multiplied by the weight for the same label I of variable Vf Su = J2jeA(vi) wij x hj where Wij is the edge weight obtained in Equation 1 and vertex Vi has |A(t>j)| adjacent vertices. In our version of the algorithm for coreference resolution A(vi) is the list of adjacent vertices of v% but only consid­ering the ones with an index k < i.

The aim of the algorithm is to find a weighted labeling such that global consistency is maxi­mized. Maximizing global consistency is defined

Figure 3: Representation of Relax. The vertices represent­ing mentions are connected by weighted edges . Each ver­tex has a vector hl of probabilities to belong to different par­titions. The figure shows h, h and h.

as maximizing the average support for each vari­able. The final partitioning is directly obtained from the weighted labeling H assigning to each variable the label with maximum probability.

The pseudo-code of the relaxation algorithm can be found in Figure 2. The process updates the weights of the labels in each step until con­vergence, i.e. when no more significant changes are done in an iteration. Finally, the assigned label for a variable is the one with the highest weight. Figure 3 shows an example of the process.

The whole methodology of RelaxCor including the resolution algorithm and the training process is totally independent of the language of the docu­ment. The only parts that need few adjustments are the preprocess and the set of feature functions. In most cases, the modifications in the feature func­tions are just for the different format of the data for different languages rather than for specific lan­guage issues. Moreover, given that the task in­cludes many information about the mentions of the documents such as part of speech, syntactic depen­dency, head and semantic role, no preprocess has been needed.

One of the problems we have found adapting the system to the task corpora was the large amount of available data. As described in Section 2.2, the training process generates a feature vector for each pair of mentions into a document for all the documents of the training data set. However, the great number of training documents and their length overwhelmed the software that learns the constraints. In order to reduce the amount of pair examples, we run a clustering process to reduce the number of negative examples using the posi­tive examples as the centroids. Note that negative examples are near 94% of the training examples, and many of them are repeated. For each positive example (a corefering pair of mentions), only the negative examples with distance less than a thresh­old d are included in the final training data. The distance is computed as the number of different values inside the feature vector. After some exper­iments over development data, the value of d was assigned to 3. Thus, the negative examples were discarded when they have more than three features different than any positive example.

Our results for the development data set are shown in Table 1.

Results of RelaxCor for the test data set are shown in Table 2. One of the characteristics of the sys­tem is that the resolution process always takes into account the whole set of mentions and avoids any possible pair-linkage contradiction as well as forces transitivity. Therefore, the system favors the precision, which results on high scores with metrics CEAF and B. However, the system is penalized with the metrics based on pair-linkage, specially with MUC. Although RelaxCor has the highest precision scores even for MUC, the recall is low enough to finally obtain low scores for Fi.

Regarding the test scores of the task comparing with the other participants (Recasens et al., 2010), RelaxCor obtains the best performances for Cata-

Table 2: Results of the task lan (CEAF and B ), English (closed: CEAF and B; open: B) and Spanish (B). Moreover, Relax­Cor is the most precise system for all the metrics in all the languages except for CEAF in English-open and Spanish. This confirms the robustness of the results of RelaxCor but also remarks that more knowledge or more information is needed to in­crease the recall of the system without loosing this precision

The incorporation of WordNet to the English run is the only difference between English-open and English-closed. The scores are slightly higher when using WordNet but not significant. Analyz­ing the MUC scores, note that the recall is im­proved, while precision decreases a little which corresponds with the information and the noise that WordNet typically provides.

The results for the test and development are very similar as expected, except the Spanish (es) ones. The recall considerably falls from develop­ment to test. It is clearly shown in the MUC recall and also is indirectly affecting on the other scores.