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Polynomial Precision Dependence Solutions to Alignment Research Center Matrix Completion Problems

We present solutions to the matrix completion problems proposed by the Alignment Research Center that have a polynomial dependence on the precision $\varepsilon$. The motivation for these problems is to enable efficient computation of heuristic estimators to formally evaluate and reason about different quantities of deep neural networks in the interest of AI alignment. Our solutions involve reframing the matrix completion problems as a semidefinite program (SDP) and using recent advances in spectral bundle methods for fast, efficient, and scalable SDP solving.