Empirical Bayes 1-bit matrix completion
Authors: Takeru Matsuda
Summary
The paper develops a method for predicting missing entries in binary matrices (like user-item ratings that are 0 or 1), a problem called 1-bit matrix completion. The approach is inspired by the James-Stein estimator and shrinks the singular values of the matrix toward zero, exploiting the fact that many real binary matrices are approximately low-rank. They connect this to multidimensional item response theory from psychometrics. Simulations and real data show the method balances predictive accuracy, calibration (whether predicted probabilities match observed frequencies), and speed better than existing methods.
Main takeaways:
- Tackles binary matrix completion (predicting unobserved 0/1 entries) for applications like recommendation systems
- Uses empirical Bayes shrinkage of singular values, generalizing the James-Stein estimator to matrices
- Exploits low-rank structure and connects to item response theory
- Achieves good trade-offs between prediction accuracy, uncertainty quantification (calibration), and computational efficiency
- Validated on simulations and real datasets
Relevance
Not connected to my work on language model personas or behavior steering—this is a statistical method for collaborative filtering and matrix completion.
Abstract
arXiv:2605.09509v1 Announce Type: new Abstract: The problem of predicting unobserved entries in a binary matrix, known as 1-bit matrix completion, has found diverse applications in fields such as recommendation systems. In this study, we develop an empirical Bayes method for 1-bit matrix completion motivated by the Efron--Morris estimator, a matrix generalization of the James--Stein estimator that shrinks singular values toward zero. The proposed method exploits the underlying low-rank structure of binary matrices, drawing parallels with multidimensional item response theory. Simulation studies and real-data applications demonstrate that the proposed method achieves a superior balance of predictive accuracy, calibration reliability (uncertainty quantification), and computational efficiency compared to existing methods.