If my entire training set of documents is class A, how can I use TF-IDF to find other documents of class A?
What I think you have is an unsupervised learning application.  Clustering.  Using the combined X & Y dataset, generate clusters.  Then overlay the X boundary; the boundary that contains all X samples.  All items from Y in the X boundary can be considered X.  And the X-ness of a given sample from Y is the distance from the X cluster centroid.  Something like that.
I have a collection X of documents, all of which are of class A (the only class in which I'm interested or know anything about). I also have a much larger collection Y of documents that I know nothing about. The documents in X and Y come from the same source and have similar formats and somewhat similar subject matters. I'd like to use the TF-IDF feature vectors of the documents in X to find the documents in Y that are most likely to be of class A.
In the past, I've used TF-IDF feature vectors to build naive Bayes classifiers, but in these situations, my training set X consisted of documents of many classes, and my objective was to classify each document in Y as one of the classes seen in X.
This seems like a different situation. Here, my entire training set has the same class (I have no documents that I know are not of class A), and I'm only interested in determining if documents in Y are or are not of that class.
A classifier seems like the wrong route, but I'm not sure what the best next step is. Is there a different algorithm that can use that TF-IDF matrix to determine the likelihood that a document is of the same class?
FYI, I'm using scikit-learn in Python 2.7, which obviously made computing the TF-IDF matrix of X (and Y) simple.