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Further \(\exists {\mathbb {R}}\)-Complete Problems with PSD Matrix Factorizations

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Abstract

Let A be an \(m\times n\) matrix with nonnegative real entries. The psd rank of A is the smallest k for which there exist two families \((P_1,\ldots ,P_m)\) and \((Q_1,\ldots ,Q_n)\) of positive semidefinite Hermitian \(k\times k\) matrices such that \(A(i|j)={\text {tr}}(P_i Q_j)\) for all i, j. Several questions on the algorithmic complexity of related matrix invariants were posed in recent literature: (i) by Stark (for the psd rank as defined above), (ii) by Goucha, Gouveia (for phaseless rank, which appears if the matrices \(P_i\) and \(Q_j\) are required to be of rank one in the above definition), (iii) by Gribling, de Laat, Laurent (for cpsd rank, which corresponds to the situation when A is symmetric and \(P_i=Q_i\) for all i). We solve these questions by proving that the decision versions of the corresponding invariants are \(\exists {\mathbb {R}}\)-complete. In addition, we give a polynomial time recognition algorithm for matrices of bounded cpsd rank.

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    Yaroslav Shitov

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Shitov, Y. Further \(\exists {\mathbb {R}}\)-Complete Problems with PSD Matrix Factorizations. Found Comput Math (2023). https://doi.org/10.1007/s10208-023-09610-1

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