Low-degree polynomial conjecture for indistinguishability in the Gaussian mixture testing model

Establish that, in the hypothesis testing problem with H0: X_i i.i.d. N(0, I_p) and H1: X_i i.i.d. \{ \varpi/(1+\varpi) N(\mu, I_p) + 1/(1+\varpi) N(-\varpi\mu, I_p) \}, if the degree-D orthogonal projection of the likelihood ratio under the null satisfies E_0[ \Lambda_{(D)}^2 ] = 1 + o(1) for D = (log p)^{1.01}, then no polynomial-time algorithm can distinguish the null and alternative distributions; i.e., prove the stated low-degree polynomial conjecture for this Gaussian mixture testing setting.

Background

To derive computational minimax lower bounds, the paper invokes the low-degree polynomial conjecture, a widely used but unproven hypothesis linking the failure of low-degree polynomial tests (measured via the second moment of the degree-D projection of the likelihood ratio) to the impossibility of polynomial-time algorithms. The authors define the null and alternative distributions for their Gaussian mixture testing problem and state the conjecture in this context.

A proof of this conjecture in the specified model would rigorously justify computational lower bounds used to delineate the statistical–computational gap for clustering and signal recovery, strengthening the theoretical foundation of the paper’s CMLB results.