  Many problems in machine learning can be reduced to learning a low-rank
positive semidefinite matrix (denoted as $Z$), which encounters semidefinite
program (SDP). Existing SDP solvers by classical convex optimization are
expensive to solve large-scale problems. Employing the low rank of solution,
Burer-Monteiro's method reformulated SDP as a nonconvex problem via the
$quadratic$ factorization ($Z$ as $XX^\top$). However, this would lose the
structure of problem in optimization. In this paper, we propose to convert SDP
into a biconvex problem via the $bilinear$ factorization ($Z$ as $XY^\top$),
and while adding the term $\frac{\gamma}{2}||X-Y||_F^2$ to penalize the
difference of $X$ and $Y$. Thus, the biconvex structure (w.r.t. $X$ and $Y$)
can be exploited naturally in optimization. As a theoretical result, we provide
a bound to the penalty parameter $\gamma$ under the assumption of $L$-Lipschitz
smoothness and $\sigma $-strongly biconvexity, such that, at stationary points,
the proposed bilinear factorization is equivalent to Burer-Monteiro's
factorization when the bound is arrived, that is
$\gamma>\frac{1}{4}(L-\sigma)_+$. Our proposal opens up a new way to surrogate
SDP by biconvex program. Experiments on two SDP-related applications
demonstrate that the proposed method is effective as the state-of-the-art.
