  We study a general matrix optimization problem with a fixed-rank positive
semidefinite (PSD) constraint. We perform the Burer-Monteiro factorization and
consider a particular Riemannian quotient geometry in a search space that has a
total space equipped with the Euclidean metric. When the original objective f
satisfies standard restricted strong convexity and smoothness properties, we
characterize the global landscape of the factorized objective under the
Riemannian quotient geometry. We show the entire search space can be divided
into three regions: (R1) the region near the target parameter of interest,
where the factorized objective is geodesically strongly convex and smooth; (R2)
the region containing neighborhoods of all strict saddle points; (R3) the
remaining regions, where the factorized objective has a large gradient. To our
best knowledge, this is the first global landscape analysis of the
Burer-Monteiro factorized objective under the Riemannian quotient geometry. Our
results provide a fully geometric explanation for the superior performance of
vanilla gradient descent under the Burer-Monteiro factorization. When f
satisfies a weaker restricted strict convexity property, we show there exists a
neighborhood near local minimizers such that the factorized objective is
geodesically convex. To prove our results we provide a comprehensive landscape
analysis of a matrix factorization problem with a least squares objective,
which serves as a critical bridge. Our conclusions are also based on a result
of independent interest stating that the geodesic ball centered at Y with a
radius 1/3 of the least singular value of Y is a geodesically convex set under
the Riemannian quotient geometry, which as a corollary, also implies a
quantitative bound of the convexity radius in the Bures-Wasserstein space. The
convexity radius obtained is sharp up to constants.
