  A rank-$r$ matrix $X \in \mathbb{R}^{m \times n}$ can be written as a product
$U V^\top$, where $U \in \mathbb{R}^{m \times r}$ and $V \in \mathbb{R}^{n
\times r}$. 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 rank-$r$ matrices is modeled via the factorization $UV^\top$. 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: $f(UV^\top)$ becomes a non-convex function w.r.t.
$U$ and $V$.
  We study such parameterization for optimization of generic convex objectives
$f$, and focus on first-order, gradient descent algorithmic solutions. We
propose the Bi-Factored Gradient Descent (BFGD) algorithm, an efficient
first-order method that operates on the $U, V$ factors. We show that when $f$
is (restricted) smooth, BFGD has local sublinear convergence, and linear
convergence when $f$ is both (restricted) smooth and (restricted) strongly
convex. For several key applications, we provide simple and efficient
initialization schemes that provide approximate solutions good enough for the
above convergence results to hold.
