Extend AMP–low-degree optimality beyond constant degree

Extend the result that, in the spiked Wigner model, the exact mean squared error achieved by approximate message passing is optimal among constant-degree polynomials by proving the same optimality for polynomials of degree that grows with n (e.g., logarithmic degree), thereby matching replica predictions at higher degrees.

Background

Physics-based methods (AMP, replica) predict sharp phase transitions and exact asymptotic mean squared errors in spiked models. Recent work proves that, in the spiked Wigner setting, AMP’s MSE is optimal among constant-degree polynomials.

The survey notes that extending this optimality to higher-degree polynomials is open, which would solidify the connection between AMP’s predictions and low-degree algorithmic limits.