论文标题
具有线性参数化因子的非convex矩阵完成
Nonconvex Matrix Completion with Linearly Parameterized Factors
论文作者
论文摘要
矩阵完成的技术旨在通过一小部分观察到的数据矩阵中的大部分丢失条目进行估算。在实际上，通常采用包括协作过滤，以前的信息和特殊结构，以提高矩阵完成的准确性。在本文中，我们提出了一个统一的非凸优化框架，以实现线性参数化因子的矩阵完成。特别是，通过引入称为相关参数分解的条件，我们可以通过建立任何局部最小值导致的低排名估计的统一上限来对非convex物镜进行统一的几何分析。也许令人惊讶的是，相关参数分解的条件适用于重要示例，包括子空间约束的矩阵完成和偏斜 - 对称矩阵的完成。我们统一的非凸优化方法的有效性也通过广泛的数值模拟在经验上说明。
Techniques of matrix completion aim to impute a large portion of missing entries in a data matrix through a small portion of observed ones. In practice including collaborative filtering, prior information and special structures are usually employed in order to improve the accuracy of matrix completion. In this paper, we propose a unified nonconvex optimization framework for matrix completion with linearly parameterized factors. In particular, by introducing a condition referred to as Correlated Parametric Factorization, we can conduct a unified geometric analysis for the nonconvex objective by establishing uniform upper bounds for low-rank estimation resulting from any local minimum. Perhaps surprisingly, the condition of Correlated Parametric Factorization holds for important examples including subspace-constrained matrix completion and skew-symmetric matrix completion. The effectiveness of our unified nonconvex optimization method is also empirically illustrated by extensive numerical simulations.
