Learning from Totally Unpaired Data via Minimum Conditional Entropy Coupling

Consider an "unsupervised classification" setting, where the goal is to learn a predictor of the categorical class variable X using the categorical feature variables Y1,ldots,Ym, but we can only train the predictor on m+1 totally unpaired datasets, where each dataset only consists of unordered samples of one of the variables X,Y1,ldots,Ym. Although this problem is highly underdetermined, we demonstrate that, under the assumption that a predictor with small cross entropy loss exists, we can find a good predictor by coupling the marginal distributions of X,Y1,ldots,Ym in a manner that minimizes the conditional entropy H(X|Y1,ldots,Ym). This problem, which we call minimum conditional entropy coupling (MCEC), is a generalization of the widely studied minimum entropy coupling problem (where m=1). Although the MCEC problem is NP-hard, we give an algorithm with a time complexity that is polynomial in N, which gives a coupling with conditional entropy within O(log log N) from the optimum, where N is the dimension of the output coupling. We also evaluate the performance of the MCEC predictor on synthetic and real-world datasets.