Mutual Predictability Decomposition: Learning Interpretable Cross-Set Structure via Bi-Directional Prediction

Understanding relationships between variable sets is fundamental in machine learning, particularly in medical data analysis. Existing methods typically provide only coarse shared/private partitioning of variables, limiting their ability to reveal finer-grained predictive structures across views. We propose Mutual Predictability Decomposition (MPD), a framework that decomposes the variables in each view into three subsets: mutually predictable variables that can be inferred across views, auxiliary variables that are themselves not cross-view predictable but are necessary for predicting the mutually predictable subset, and unpredictable variables that do not contribute to cross-view predictability. MPD is formulated as a coupled bi-directional prediction problem with sparse variable assignment, and optimized through a differentiable relaxation implemented using gated neural networks. Experiments on synthetic and real-world medical datasets demonstrate that MPD recovers interpretable cross-view structures, improves cross-view prediction, and provides more informative decompositions than existing relationship analysis methods. Code will be available on GitHub upon acceptance.