  This paper considers the minimization of a general objective function $f(X)$
over the set of rectangular $n\times m$ matrices that have rank at most $r$. To
reduce the computational burden, we factorize the variable $X$ into a product
of two smaller matrices and optimize over these two matrices instead of $X$.
Despite the resulting nonconvexity, recent studies in matrix completion and
sensing have shown that the factored problem has no spurious local minima and
obeys the so-called strict saddle property (the function has a directional
negative curvature at all critical points but local minima). We analyze the
global geometry for a general and yet well-conditioned objective function
$f(X)$ whose restricted strong convexity and restricted strong smoothness
constants are comparable. In particular, we show that the reformulated
objective function has no spurious local minima and obeys the strict saddle
property. These geometric properties imply that a number of iterative
optimization algorithms (such as gradient descent) can provably solve the
factored problem with global convergence.
