一种具有自适应后验收缩性质的贝叶斯稀疏因子模型
A Bayesian sparse factor model with adaptive posterior concentration
AI总结：
本文提出一种新的高维稀疏因子模型贝叶斯推断方法，通过引入稀疏水平与因子维数之间的依赖关系，在保持计算可处理性的同时实现自适应后验收缩，并证明其在因子维数检测和协方差矩阵估计上具有最优或近似最优的理论性质与数值优势。
摘要：
在本文中，我们针对高维稀疏因子模型提出了一种新的贝叶斯推断方法，该方法允许同时推断因子维数和载荷矩阵的稀疏结构。其新颖之处在于引入稀疏水平与因子维数之间的某种依赖关系，从而在保持计算可处理性的同时实现自适应后验收缩。我们证明后验分布渐近地集中在真实因子维数上，更重要的是，这种后验相合性对真实载荷矩阵的稀疏水平和噪声方差具有自适应性。我们还证明，所提出的贝叶斯模型在比文献中已有结果更一般的情形下达到了因子维数的最优检测速率。此外，我们得到了协方差矩阵的近似最优后验收缩速率。数值研究表明，与其他竞争方法相比，所提出的方法具有优越性。
英文摘要：
In this paper, we propose a new Bayesian inference method for a high-dimensional sparse factor model that allows both the factor dimensionality and the sparse structure of the loading matrix to be inferred. The novelty is to introduce a certain dependence between the sparsity level and the factor dimensionality, which leads to adaptive posterior concentration while keeping computational tractability. We show that the posterior distribution asymptotically concentrates on the true factor dimensionality, and more importantly, this posterior consistency is adaptive to the sparsity level of the true loading matrix and the noise variance. We also prove that the proposed Bayesian model attains the optimal detection rate of the factor dimensionality in a more general situation than those found in the literature. Moreover, we obtain a near-optimal posterior concentration rate of the covariance matrix. Numerical studies are conducted and show the superiority of the proposed method compared with other competitors.
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