Low-degree upper bound at super-logarithmic degree in planted submatrix

Show that, in the planted submatrix model (where an unknown k×k submatrix has elevated mean controlled by a signal-to-noise parameter λ), the degree-D minimum mean squared error is small for D≈λ^{-2}, establishing a matching low-degree upper bound at super-logarithmic degree that tracks super-polynomial runtime predictions.

Background

The survey highlights a notable gap: low-degree upper bounds that match super-polynomial runtime tradeoffs are largely missing. In particular, the planted submatrix model is a canonical setting where super-logarithmic degree should correspond to super-polynomial time.

Proving that the degree-D MMSE is small for D≈λ^{-2} would provide a concrete upper bound aligning the low-degree framework with known runtime–SNR tradeoffs in this model.