Low-rank matrix recovery allows a low-rank matrix to be reconstructed when only a fraction of its elements is available. In this paper, an approximate Bayesian approach to low-rank matrix recovery is developed and its potential benefit for an application in metrology explored. The approach extends a recently proposed Bayesian low-rank matrix recovery procedure by utilizing a Gaussian Markov random field (GMRF) prior. The GMRF prior accounts for spatial smoothness, which is relevant for applications such as quantitative magnetic resonance imaging and nano Fourier transform infrared (FTIR) spectroscopy. The approach proposed here is automatic in that its hyperparameters are estimated from the data. Application to nano-FTIR spectroscopy demonstrates that the effort required to perform experiments in the time-consuming measurement of multi-dimensional data can be reduced significantly. Software for the proposed approach is available upon request.

低秩矩阵恢复允许仅在其元素的一小部分可用时重建低秩矩阵。在本文中，发展了一种近似的贝叶斯方法进行低秩矩阵恢复，并探讨了其在计量学中的潜在优势。该方法通过利用先前的高斯马尔可夫随机场（GMRF）扩展了最近提出的贝叶斯低秩矩阵恢复程序。GMRF先验考虑了空间平滑度，这与诸如定量磁共振成像和纳米傅里叶变换红外（FTIR）光谱学等应用有关。这里提出的方法是自动的，因为它的超参数是根据数据估算的。在纳米FTIR光谱学中的应用表明，在耗时的多维数据测量中进行实验所需的工作量可以大大减少。可根据要求提供用于建议方法的软件。