Flexible Low-Rank Statistical Modeling with Missing Data and Side Information
Flexible Low-Rank Statistical Modeling with Missing Data and Side Information
复制标题
DOI：
10.1214/18-sts642
复制
发表时间：
2018-05
影响因子：
5.7
通讯作者：
William Fithian;R. Mazumder
中科院分区：
文献类型：
--
作者：
William Fithian;R. Mazumder
我们探索了一个一般的统计框架，矩阵值数据的低秩建模，基于凸优化与广义核范数惩罚。我们研究了几个相关的问题：通常的低秩矩阵完成问题与灵活的损失函数所产生的广义线性模型;降秩回归和多任务学习;和泛化的行和列的边信息是可用的，在功能或平滑内核的形式的问题。我们表明，我们的方法包括最大后验概率估计所产生的贝叶斯分层模型与潜在因素，并讨论了在矩阵完成的背景下的丢失数据机制的后果。虽然上述问题可以很自然地作为秩约束优化问题，这是非凸的，计算困难，我们展示了如何放松它们通过广义核范数正则化，以获得凸优化问题。我们讨论的算法从现代凸优化方法的启发，以解决这些大规模的凸优化计算任务。最后，我们说明了我们的灵活的方法在功能数据重建和生态物种分布建模中出现的问题。
We explore a general statistical framework for low-rank modeling of matrix-valued data, based on convex optimization with a generalized nuclear norm penalty. We study several related problems: the usual low-rank matrix completion problem with flexible loss functions arising from generalized linear models; reduced-rank regression and multi-task learning; and generalizations of both problems where side information about rows and columns is available, in the form of features or smoothing kernels. We show that our approach encompasses maximum a posteriori estimation arising from Bayesian hierarchical modeling with latent factors, and discuss ramifications of the missing-data mechanism in the context of matrix completion. While the above problems can be naturally posed as rank-constrained optimization problems, which are nonconvex and computationally difficult, we show how to relax them via generalized nuclear norm regularization to obtain convex optimization problems. We discuss algorithms drawing inspiration from modern convex optimization methods to address these large scale convex optimization computational tasks. Finally, we illustrate our flexible approach in problems arising in functional data reconstruction and ecological species distribution modeling.