The problem of estimating a matrix based on a set of its observed entries is commonly referred to as the matrix completion problem. In this work, we specifically address the scenario of binary observations, often termed as 1-bit matrix completion. While numerous studies have explored Bayesian and frequentist methods for real-value matrix completion, there has been a lack of theoretical exploration regarding Bayesian approaches in 1-bit matrix completion. We tackle this gap by considering a general, non-uniform sampling scheme and providing theoretical assurances on the efficacy of the fractional posterior. Our contributions include obtaining concentration results for the fractional posterior and demonstrating its effectiveness in recovering the underlying parameter matrix. We accomplish this using two distinct types of prior distributions: low-rank factorization priors and a spectral scaled Student prior, with the latter requiring fewer assumptions. Importantly, our results exhibit an adaptive nature by not mandating prior knowledge of the rank of the parameter matrix. Our findings are comparable to those found in the frequentist literature, yet demand fewer restrictive assumptions.

翻译：基于观测条目估计矩阵的问题通常被称为矩阵补全问题。本文专门研究二元观测场景，即通常所说的1比特矩阵补全。尽管已有大量研究探讨了实值矩阵补全的贝叶斯和频率学派方法，但关于1比特矩阵补全中贝叶斯方法的理论探索尚显不足。我们通过考虑一种通用的非均匀采样方案来弥补这一空白，并为分数后验的有效性提供理论保证。我们的贡献包括：获得分数后验的集中性结果，并证明其在恢复底层参数矩阵方面的有效性。我们通过两种不同类型的先验分布实现这一目标：低秩分解先验和谱尺度化学生先验，后者所需假设更少。重要的是，我们的结果具有自适应性，无需事先了解参数矩阵的秩。所得结论可与频率学派文献中的结果相媲美，但所需限制性假设更少。