Rule out low-degree polynomial threshold function tests

Prove unconditional lower bounds showing that, for high-dimensional average-case detection tasks between a planted distribution and a product-measure null distribution, any test that thresholds the output of a degree-D polynomial (i.e., a polynomial threshold function) fails to achieve strong or weak detection as n→∞, thereby ruling out polynomial threshold function tests as a class of low-degree algorithms.

Background

The survey formalizes success for low-degree detection via separation—controlling the first two moments of a polynomial under planted and null distributions. Current lower bounds rule out separation but do not directly exclude tests that threshold a low-degree polynomial (polynomial threshold functions, or PTFs).

The author emphasizes that while existing results show the standard Chebyshev-based analysis of PTFs fails, they do not formally preclude PTFs themselves. Establishing lower bounds that explicitly rule out PTF tests would strengthen the framework and align it with commonly used testing procedures.