  Given a matrix $X\in \mathbb{R}^{m\times n}_+$ with non-negative entries, the
cone factorization problem over a cone $\mathcal{K}\subseteq \mathbb{R}^k$
concerns computing $\{ a_1,\ldots, a_{m} \} \subseteq \mathcal{K}$ and $\{
b_1,\ldots, b_{n} \} \subseteq~\mathcal{K}^*$ belonging to its dual so that
$X_{ij} = \langle a_i, b_j \rangle$ for all $i\in [m], j\in [n]$. Cone
factorizations are fundamental to mathematical optimization as they allow us to
express convex bodies as feasible regions of linear conic programs. In this
paper, we introduce and analyze the symmetric-cone multiplicative update (SCMU)
algorithm for computing cone factorizations when $\mathcal{K}$ is symmetric;
i.e., it is self-dual and homogeneous. Symmetric cones are of central interest
in mathematical optimization as they provide a common language for studying
linear optimization over the nonnegative orthant (linear programs), over the
second-order cone (second order cone programs), and over the cone of positive
semidefinite matrices (semidefinite programs). The SCMU algorithm is
multiplicative in the sense that the iterates are updated by applying a
meticulously chosen automorphism of the cone computed using a generalization of
the geometric mean to symmetric cones. Using an extension of Lieb's concavity
theorem and von Neumann's trace inequality to symmetric cones, we show that the
squared loss objective is non-decreasing along the trajectories of the SCMU
algorithm. Specialized to the nonnegative orthant, the SCMU algorithm
corresponds to the seminal algorithm by Lee and Seung for computing Nonnegative
Matrix Factorizations.
