  This paper considers the problem of positive semidefinite factorization (PSD
factorization), a generalization of exact nonnegative matrix factorization.
Given an $m$-by-$n$ nonnegative matrix $X$ and an integer $k$, the PSD
factorization problem consists in finding, if possible, symmetric $k$-by-$k$
positive semidefinite matrices $\{A^1,...,A^m\}$ and $\{B^1,...,B^n\}$ such
that $X_{i,j}=\text{trace}(A^iB^j)$ for $i=1,...,m$, and $j=1,...n$. PSD
factorization is NP-hard. In this work, we introduce several local optimization
schemes to tackle this problem: a fast projected gradient method and two
algorithms based on the coordinate descent framework. The main application of
PSD factorization is the computation of semidefinite extensions, that is, the
representations of polyhedrons as projections of spectrahedra, for which the
matrix to be factorized is the slack matrix of the polyhedron. We compare the
performance of our algorithms on this class of problems. In particular, we
compute the PSD extensions of size $k=1+ \lceil \log_2(n) \rceil$ for the
regular $n$-gons when $n=5$, $8$ and $10$. We also show how to generalize our
algorithms to compute the square root rank (which is the size of the factors in
a PSD factorization where all factor matrices $A^i$ and $B^j$ have rank one)
and completely PSD factorizations (which is the special case where the input
matrix is symmetric and equality $A^i=B^i$ is required for all $i$).
