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Given a matrix $X\in \mathbb{R}_+^{m\times n}$ with nonnegative entries, aPositive Semidefinite (PSD) factorization of $X$ is a collection of $r \timesr$-dimensional PSD matrices $\{A_i\}$ and $\{B_j\}$ satisfying $X_{ij}=\mathrm{tr}(A_i B_j)$ for all $\ i\in [m],\ j\in [n]$. PSD factorizations arefundamentally linked to understanding the expressiveness of semidefiniteprograms as well as the power and limitations of quantum resources ininformation theory. The PSD factorization task generalizes the Non-negativeMatrix Factorization (NMF) problem where we seek a collection of$r$-dimensional nonnegative vectors $\{a_i\}$ and $\{b_j\}$ satisfying $X_{ij}=a_i^\top b_j$, for all $i\in [m],\ j\in [n]$ -- one can recover the latterproblem by choosing matrices in the PSD factorization to be diagonal. The mostwidely used algorithm for computing NMFs of a matrix is the MultiplicativeUpdate algorithm developed by Lee and Seung, in which nonnegativity of theupdates is preserved by scaling with positive diagonal matrices. In this paper,we describe a non-commutative extension of Lee-Seung's algorithm, which we callthe Matrix Multiplicative Update (MMU) algorithm, for computing PSDfactorizations. The MMU algorithm ensures that updates remain PSD by congruencescaling with the matrix geometric mean of appropriate PSD matrices, and itretains the simplicity of implementation that Lee-Seung's algorithm enjoys.Building on the Majorization-Minimization framework, we show that under ourupdate scheme the squared loss objective is non-increasing and fixed pointscorrespond to critical points. The analysis relies on Lieb's Concavity Theorem.Beyond PSD factorizations, we use the MMU algorithm as a primitive to calculateblock-diagonal PSD factorizations and tensor PSD factorizations. We demonstratethe utility of our method with experiments on real and synthetic data.

A Non-commutative Extension of Lee-Seung's Algorithm for Positive Semidefinite Factorizations