Build a binary classifier with only positive and unlabeled data
My suggestion would be to attempt to build some kind of clustering on your unlabeled data that somewhat approximates a labelled dataset. The rationale is more or less as follows:


You have some feature vector for representing your documents
Based on that feature vector, you can come up with a number of different clusterings, with either fuzzy, rough, or class-based clustering methods
Knowing what a positive example looks like, you can quickly evaluate the overall similarity of a cluster to your positive cluster
Knowing that there should really only be two clusters, you can adjust the hyperparameters on your clustering method so that the above two metrics are closer and closer to satisfaction
With the two clusters, you have what is likely a close approximation of a labelled dataset, which you can then use as a silver-standard corpus of sorts to actually train your model


Hope that makes sense, if you're specifically looking for clustering algorithms, a few that I personally enjoy that might be good in this scenario are FLAME and tsne. Alternately, looking at the spectacular gensim library in python will get you a long way toward the clustering you're looking for.

Hope that helps and makes sense, leave a comment if you've got any questions.