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.

翻译：本文提出了一种适用于高维稀疏因子模型的新型贝叶斯推断方法，该方法能够同时推断因子维度和载荷矩阵的稀疏结构。其创新之处在于引入了稀疏度与因子维度之间的特定依赖关系，从而在保持计算可行性的同时实现自适应后验集中性。我们证明了后验分布渐近地集中于真实的因子维度，更重要的是，这种后验一致性对于真实载荷矩阵的稀疏度和噪声方差具有自适应性。我们还证明了所提出的贝叶斯模型在比文献中更一般的情况下能够达到因子维度的最优检测速率。此外，我们获得了协方差矩阵的近乎最优后验集中速率。数值实验表明，与其他竞争方法相比，所提出方法具有优越性。