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We study a type of Riemannian gradient descent (RGD) algorithm, designedthrough Riemannian preconditioning, for optimization on $\mathcal{M}_k^{m\timesn}$ -- the set of $m\times n$ real matrices with a fixed rank $k$. Our analysisis based on a quotient geometric view of $\mathcal{M}_k^{m\times n}$: byidentifying 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 fullcolumn rank via matrix factorization, we find an explicit form for the updaterule of the RGD algorithm, which leads to a novel approach to analysing theirconvergence behavior in rank-constrained optimization. We then deduce someinteresting properties that reflect how RGD distinguishes from other matrixfactorization algorithms such as those based on the Euclidean geometry. Inparticular, we show that the RGD algorithm are not only faster than Euclideangradient descent but also do not rely on the balancing technique to ensure itsefficiency while the latter does. Starting from the novel results, we furthershow that this RGD algorithm is guaranteed to solve matrix sensing and matrixcompletion problems with linear convergence rate, under mild conditions relatedto the restricted positive definiteness property. Numerical experiments onmatrix sensing and completion are provided to demonstrate these properties.

On the analysis of optimization with fixed-rank matrices: a quotient geometric view