Matrix completion aims to estimate missing entries in a data matrix, using the assumption of a low-complexity structure (e.g., low rank) so that imputation is possible. While many effective estimation algorithms exist in the literature, uncertainty quantification for this problem has proved to be challenging, and existing methods are extremely sensitive to model misspecification. In this work, we propose a distribution-free method for predictive inference in the matrix completion problem. Our method adapts the framework of conformal prediction, which provides confidence intervals with guaranteed distribution-free validity in the setting of regression, to the problem of matrix completion. Our resulting method, conformalized matrix completion (cmc), offers provable predictive coverage regardless of the accuracy of the low-rank model. Empirical results on simulated and real data demonstrate that cmc is robust to model misspecification while matching the performance of existing model-based methods when the model is correct.

翻译：矩阵补全旨在利用低复杂度结构（如低秩性）假设估计数据矩阵中的缺失条目，从而完成插补。尽管文献中存在多种有效估计算法，但该问题的不确定性量化仍具挑战性，且现有方法对模型误设极度敏感。本文提出一种用于矩阵补全问题预测推断的无分布方法。该方法将共形预测框架（在回归场景中提供具有保障性分布自由有效性的置信区间）适配至矩阵补全问题。由此产生的共形化矩阵补全（CMC）方法，无论低秩模型精度如何，均能提供可证明的预测覆盖。模拟与真实数据实证结果表明，CMC方法对模型误设具有鲁棒性，同时在模型正确时能与现有基于模型方法的性能相匹配。