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A rank-$r$ matrix $X  mathbb{R}^{m times n}$ can be written as a product $U V^top$, where $U  mathbb{R}^{m times r}$ and $V  mathbb{R}^{n times r}$. One could exploit this observation in optimization: e.g., consider the minimization of a convex function $f(X)$ over rank-$r$ matrices, where the set of low-rank matrices is modeled via $UV^top$. Though such parameterization reduces the number of variables and is more computationally efficient (of particular interest is the case $r ll min{m, n}$), it comes at a cost: $f(U V^top)$ becomes a nonconvex function w.r.t. $U$ and $V$. We study such parameterization on generic convex objectives $f$ and focus on first-order, gradient descent algorithms. We propose the bifactored gradient descent (textttBFGD) algorithm, an efficient first-order method that operates directly on the $U, V$ factors. We show that when $f$ is (restricted) smooth, textttBFGD has local sublinear convergence; when $f$ is both (restricted) smooth and (restricted) strongl...

Finding Low-Rank Solutions via Nonconvex Matrix Factorization, Efficiently and Provably