Lower bounds for constrained moment-matching in the interpolation regime

Establish the conjectured lower bounds for the constrained moment-matching problem M_p(ε,L) that underlie interpolation-regime lower bounds for tolerant testing under non-smooth ℓp norms. Specifically, prove that for 0<ε≲1/L, one has M_p(ε,L)^{1/p} ≳ ε^{p/2} L^{1−p/2} when 1≤p<2, and M_p(ε,L)^{1/p} ≳ ε^{2k/p} L^{1−2k/p} when 2k<p<2(k+1) with k∈Z+. Here M_p(ε,L) denotes the supremum of E_{π1}|v|^p over distributions π0,π1 on [−1,1] that match the first L moments and satisfy E_{π0}|v|^p ≤ ε^p.

Background

To obtain minimax lower bounds for tolerant testing rates, the paper uses a reduction to moment-matching problems. For the interpolation regime in non-smooth ℓp settings, this requires constrained moment-matching where the mean under the null is restricted. The authors formulate explicit conjectured bounds for the constrained optimization M_p(ε,L), mirroring techniques that succeeded in the discrete distribution setting and linking to best polynomial approximation via duality.

Proving these bounds would yield sharp lower bounds for the interpolation regime and, combined with the established upper bounds, would settle the minimax rates (up to polylog factors) in the non-smooth ℓp tolerant testing problem.