Low rank inference on matrices is widely conducted by optimizing a cost function augmented with a penalty proportional to the nuclear norm $\Vert \cdot \Vert_*$. However, despite the assortment of computational methods for such problems, there is a surprising lack of understanding of the underlying probability distributions being referred to. In this article, we study the distribution with density $f(X)\propto e^{-\lambda\Vert X\Vert_*}$, finding many of its fundamental attributes to be analytically tractable via differential geometry. We use these facts to design an improved MCMC algorithm for low rank Bayesian inference as well as to learn the penalty parameter $\lambda$, obviating the need for hyperparameter tuning when this is difficult or impossible. Finally, we deploy these to improve the accuracy and efficiency of low rank Bayesian matrix denoising and completion algorithms in numerical experiments.

翻译：矩阵的低秩推断通常通过优化一个成本函数并加上与核范数 $\Vert \cdot \Vert_*$ 成比例的惩罚项来实现。然而，尽管针对此类问题存在多种计算方法，人们对所涉及的基础概率分布却惊人地缺乏理解。在本文中，我们研究了密度为 $f(X)\propto e^{-\lambda\Vert X\Vert_*}$ 的分布，发现通过微分几何方法可以解析地处理其许多基本属性。我们利用这些事实设计了一种改进的MCMC算法用于低秩贝叶斯推断，并学习惩罚参数 $\lambda$，从而在超参数调优困难或不可能时避免其需求。最后，我们通过数值实验将这些方法应用于提升低秩贝叶斯矩阵去噪与补全算法的准确性和效率。