This paper studies the open problem of conformalized entry prediction in a row/column-exchangeable matrix. The matrix setting presents novel and unique challenges, but there exists little work on this interesting topic. We meticulously define the problem, differentiate it from closely related problems, and rigorously delineate the boundary between achievable and impossible goals. We then propose two practical algorithms. The first method provides a fast emulation of the full conformal prediction, while the second method leverages the technique of algorithmic stability for acceleration. Both methods are computationally efficient and can effectively safeguard coverage validity in presence of arbitrary missing pattern. Further, we quantify the impact of missingness on prediction accuracy and establish fundamental limit results. Empirical evidence from synthetic and real-world data sets corroborates the superior performance of our proposed methods.

翻译：本文研究了行/列可交换矩阵中经过保形化处理的条目预测这一开放性问题。矩阵设定带来了全新且独特的挑战，而目前关于这一有趣课题的研究甚少。我们严谨地定义了该问题，将其与密切相关的其他问题加以区分，并细致界定了可实现目标与不可能目标之间的边界。随后我们提出了两种实用算法：第一种方法可快速模拟完整保形预测，第二种方法则利用算法稳定性技术进行加速。两种方法均具有计算高效性，且能在任意缺失模式下有效保障覆盖有效性。此外，我们量化了缺失对预测精度的影响，并建立了基础极限结果。来自合成数据集与真实数据集的实证证据一致验证了我们所提出方法的优越性能。