PAC-Bayesian Analysis of Co-clustering and Beyond
Yevgeny Seldin and Naftali Tishby
Journal of Machine Learning Research
We derive PAC-Bayesian generalization bounds for supervised and unsupervised learning models based on clustering, such as co-clustering, matrix tri-factorization, graphical models, graph clustering, and pairwise clustering. We begin with the analysis of co-clustering, which is a widely used approach to the analysis of data matrices. We distinguish among two tasks in matrix data analysis: discriminative prediction of the missing entries in data matrices and estimation of the joint probability distribution of row and column variables in co-occurrence matrices. We derive PAC-Bayesian generalization bounds for the expected out-of-sample performance of co-clustering-based solutions for these two tasks. The analysis yields regularization terms that were absent in the previous formulations of co-clustering. The bounds suggest that the expected performance of co-clustering is governed by a trade-off between its empirical performance and the mutual information preserved by the cluster variables on row and column IDs. We derive an iterative projection algorithm for finding a local optimum of this trade-off for discriminative prediction tasks. This algorithm achieved state-of-the-art performance in the MovieLens collaborative filtering task. Our co-clustering model can also be seen as matrix tri-factorization and the results provide generalization bounds, regularization terms, and new algorithms for this form of matrix factorization.
The analysis of co-clustering is extended to tree-shaped graphical models, which can be used to analyze high dimensional tensors. According to the bounds, the generalization abilities of tree-shaped graphical models depend on a trade-off between their empirical data fit and the mutual information that is propagated up the tree levels.
We also formulate weighted graph clustering as a prediction problem: given a subset of edge weights we analyze the ability of graph clustering to predict the remaining edge weights. The analysis of co-clustering easily extends to this problem and suggests that graph clustering should optimize the trade-off between empirical data fit and the mutual information that clusters preserve on graph nodes.