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The complexity of positive semidefinite matrix factorization (1606.09065v1)
Published 29 Jun 2016 in math.CO and cs.CC
Abstract: Let $A$ be a matrix with nonnegative real entries. The PSD rank of $A$ is the smallest integer $k$ for which there exist $k\times k$ real PSD matrices $B_1,\ldots,B_m$, $C_1,\ldots,C_n$ satisfying $A(i|j)=\operatorname{tr}(B_iC_j)$ for all $i,j$. This paper determines the computational complexity status of the PSD rank. Namely, we show that the problem of computing this function is polynomial-time equivalent to the existential theory of the reals.
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