We study a type of Riemannian gradient descent (RGD) algorithm, designed through Riemannian preconditioning, for optimization on $\mathcal{M}_k^{m\times n}$ -- the set of $m\times n$ real matrices with a fixed rank $k$. Our analysis is based on a quotient geometric view of $\mathcal{M}_k^{m\times n}$: by identifying this set with the quotient manifold of a two-term product space $\mathbb{R}_*^{m\times k}\times \mathbb{R}_*^{n\times k}$ of matrices with full column rank via matrix factorization, we find an explicit form for the update rule of the RGD algorithm, which leads to a novel approach to analysing their convergence behavior in rank-constrained optimization. We then deduce some interesting properties that reflect how RGD distinguishes from other matrix factorization algorithms such as those based on the Euclidean geometry. In particular, we show that the RGD algorithm is not only faster than Euclidean gradient descent but also does not rely on balancing techniques to ensure its efficiency while the latter does. We further show that this RGD algorithm is guaranteed to solve matrix sensing and matrix completion problems with linear convergence rate under the restricted positive definiteness property. Numerical experiments on matrix sensing and completion are provided to demonstrate these properties.

翻译：我们研究一种黎曼梯度下降算法，该算法通过黎曼预处理设计，用于在$\mathcal{M}_k^{m\times n}$——即$m\times n$固定秩$k$实矩阵集合——上进行优化。我们的分析基于$\mathcal{M}_k^{m\times n}$的商几何视角：通过矩阵分解将该集合与满列秩矩阵双项乘积空间$\mathbb{R}_*^{m\times k}\times \mathbb{R}_*^{n\times k}$的商流形等同，我们得到了RGD算法更新规则的显式形式，这为分析其在秩约束优化中的收敛行为提供了一种新方法。我们进而推导出若干反映RGD区别于其他矩阵分解算法（如基于欧几里得几何的算法）的有趣性质。特别地，我们证明RGD算法不仅比欧几里得梯度下降更快，而且无需依赖平衡技术来保证效率，而后者需要此类技术。我们进一步证明，在受限正定性条件下，该RGD算法能以线性收敛速率保证求解矩阵感知和矩阵补全问题。文中提供了矩阵感知与补全的数值实验以验证这些性质。