We study the problem of matrix completion in this paper. A spectral scaled Student prior is exploited to favour the underlying low-rank structure of the data matrix. We provide a thorough theoretical investigation for our approach through PAC-Bayesian bounds. More precisely, our PAC-Bayesian approach enjoys a minimax-optimal oracle inequality which guarantees that our method works well under model misspecification and under general sampling distribution. Interestingly, we also provide efficient gradient-based sampling implementations for our approach by using Langevin Monte Carlo. More specifically, we show that our algorithms are significantly faster than Gibbs sampler in this problem. To illustrate the attractive features of our inference strategy, some numerical simulations are conducted and an application to image inpainting is demonstrated.
翻译:我们研究本文中的矩阵完成问题。 一个光谱缩放学生以前被用来支持数据矩阵的基本低级结构。 我们通过PAC-Bayesian界限为我们的方法提供了彻底的理论调查。 更确切地说,我们的PAC-Bayesian方法存在微小的峰值或骨骼不平等,这保证了我们的方法在模型区分和一般抽样分布下运作良好。 有趣的是,我们还利用Langevin Monte Carlo为我们的方法提供了高效的基于梯度的抽样实施。 更具体地说,我们表明,我们的算法比Gibbs取样员在这个问题上的速度要快得多。 为了说明我们推算策略的吸引力特征,我们进行了一些数字模拟,并展示了图画的应用。