Despite the widespread application of latent factor analysis, existing methods suffer from the following weaknesses: requiring the number of factors to be known, lack of theoretical guarantees for learning the model structure, and nonidentifiability of the parameters due to rotation invariance properties of the likelihood. We address these concerns by proposing a fast correlation thresholding (CT) algorithm that simultaneously learns the number of latent factors and a rotationally identifiable model structure. Our novel approach translates this structure learning problem into the search for so-called independent maximal cliques in a thresholded correlation graph that can be easily constructed from the observed data. Our clique analysis technique scales well up to thousands of variables, while competing methods are not applicable in a reasonable amount of running time. We establish a finite-sample error bound and high-dimensional consistency for the structure learning of our method. Through a series of simulation studies and a real data example, we show that the CT algorithm is an accurate method for learning the structure of factor analysis models and is robust to violations of its assumptions.

翻译：尽管潜在因子分析应用广泛，现有方法仍存在以下不足：需要事先已知因子数量、缺乏模型结构学习的理论保证，以及由于似然函数的旋转不变性导致的参数非可识别性。针对这些问题，我们提出一种快速相关阈值化（CT）算法，该算法能够同时学习潜在因子数量及具有旋转可识别性的模型结构。我们的创新方法将这一结构学习问题转化为在由观测数据构建的相关阈值图中搜索所谓的独立极大团。该团分析技术可高效扩展至数千个变量，而现有竞争方法在合理时间内无法适用。我们为该方法的模型结构学习建立了有限样本误差界及高维一致性。通过一系列仿真实验与真实数据实例，我们验证了CT算法在学习因子分析模型结构方面的准确性，且对假设违背情况具有鲁棒性。