Factor models exhibit a fundamental tradeoff among flexibility, identifiability, and computational efficiency. Bayesian spatial factor models, in particular, face pronounced identifiability concerns and scaling difficulties. To mitigate these issues and enhance posterior inference reliability, this work proposes Projected Markov Chain Monte Carlo (ProjMC$^2$), a novel Markov Chain Monte Carlo (MCMC) sampling algorithm employing projection techniques and conditional conjugacy. ProjMC$^2$ is showcased within the context of spatial factor analysis, significantly improving posterior stability and MCMC mixing efficiency by projecting posterior sampling of latent factors onto a subspace of a scaled Stiefel manifold. Theoretical results establish convergence to the stationary distribution irrespective of initial values. Integrating this approach with scalable univariate spatial modeling strategies yields a stable, efficient, and flexible modeling and sampling methodology for large-scale spatial factor models. Simulation studies demonstrate the effectiveness and practical advantages of the proposed methods. The practical utility of the methodology is further illustrated through an analysis of spatial transcriptomic data obtained from human kidney tissues, showcasing its potential for enhancing the interpretability and robustness of spatial transcriptomics analyses.

翻译：因子模型在灵活性、可识别性与计算效率之间存在根本性的权衡。贝叶斯空间因子模型尤其面临显著的可识别性问题和扩展困难。为缓解这些问题并提升后验推断的可靠性，本研究提出了投影马尔可夫链蒙特卡洛（ProjMC$^2$），一种采用投影技术与条件共轭性的新型马尔可夫链蒙特卡洛采样算法。ProjMC$^2$在空间因子分析的背景下得以展示，通过将潜在因子的后验采样投影至缩放Stiefel流形的一个子空间，显著提升了后验稳定性与MCMC混合效率。理论结果证明，无论初始值如何，算法均能收敛至平稳分布。将此方法与可扩展的单变量空间建模策略相结合，为大规模空间因子模型提供了一种稳定、高效且灵活的建模与采样方法。模拟研究验证了所提方法的有效性与实际优势。通过对人类肾脏组织获取的空间转录组数据进行分析，进一步阐明了该方法的实用价值，展示了其在提升空间转录组学分析可解释性与鲁棒性方面的潜力。