标题： 一种低秩矩阵学习的概率基础 显示英文标题

标题： A Probabilistic Basis for Low-Rank Matrix Learning

摘要： 低秩矩阵推断通常通过优化一个成本函数来实现，该函数附加了一个与核范数成比例的惩罚项$\Vert \cdot \Vert_*$。 然而，尽管有各种计算方法用于此类问题，但对所涉及的基础概率分布的理解却令人惊讶地不足。 在本文中，我们研究了密度为$f(X)\propto e^{-\lambda\Vert X\Vert_*}$的分布，发现其许多基本属性可以通过微分几何进行解析处理。 我们利用这些事实来设计一种改进的MCMC算法，用于低秩贝叶斯推断，以及学习惩罚参数$\lambda$，从而在难以或不可能进行超参数调优时避免了这一需求。 最后，我们在数值实验中部署这些方法，以提高低秩贝叶斯矩阵去噪和补全算法的准确性和效率。 显示英文摘要

摘要： 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.