For Rayleigh-Ritz approximations, the Frobenius-norm error of a PSD cone projection is at most sqrt(2) times the residual, independent of eigenvalue gaps.
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Accuracy of approximate projection to the semidefinite cone
For Rayleigh-Ritz approximations, the Frobenius-norm error of a PSD cone projection is at most sqrt(2) times the residual, independent of eigenvalue gaps.