We propose a new geometry for fixed-rank matrices that is specifically tailored to the low-rank matrix completion problem. We tune the metric on our search space so that it suits best to minimizing the least square objective that we have and also respects the symmetry arising out of the factorized search space. At one level, it is an extension of the versatile framework of optimization on quotient manifold. At another level, our algorithm can be considered as an improved version of the state-of-the-art Gauss-Seidel algorithm namely, LMaFit. We develop necessary tools needed to perform both first-order and second-order optimization. In particular, we propose gradient descent schemes (steepest descent and conjugate gradient) and trust-region algorithms. We also show that, it is numerically cheap to perform an exact linesearch given a search direction, which makes our algorithms very competitive with the state-of-the-art on standard low-rank matrix completion instances.

本文提出了特定于低秩矩阵补全问题的新黎曼几何，利用商空间自由度来调节搜索空间的度量，开发了基于梯度下降方案的优化工具，包括陡峭下降和共轭梯度，以及信赖域算法，实现了精确线性搜索，使算法在标准低秩矩阵补全实例上具有与最先进算法相当的性能。

低秩矩阵补全的黎曼几何