Sum-of-squares barrier conjecture for random-matrix large value certification

Demonstrate that for random T×N matrices with i.i.d. N(0,1) entries, in the regime σ ≤ 1 − α/4 (with T = N^α and 1 < α < 2), any constant-degree sum-of-squares method cannot certify large-value bounds stronger than the operator-norm baseline, concretely failing to prove |W| ≲ N^{α + 1 − 2σ − c} for any c > 0.

Background

This conjecture, formulated in Diakonikolas–Hopkins–Pensia–Tiegel building on Mao–Wein, posits intrinsic limits of SOS in certifying sparse large-value phenomena for random matrices, aligning with broader average-case complexity barriers observed in planted clique and random tensor problems.

It delineates the parameter regime where polynomial-time certificate methods should match the simple operator norm argument, implying that stronger certificates (and proofs) likely require fundamentally different techniques.