For a known finite-dimensional model subspace, classical Gaussian regression theory guarantees that least squares eventually attains the parametric rate, with prediction error scaling as the subspace dimension divided by the sample size. Yet subspaces with the same number of unknown coefficients can require very different sample sizes to reach this regime. For subspaces of polynomials on the Boolean cube, a fundamental question is how restrictions on degree and ambient dimension jointly determine this threshold. Our main result sharply characterizes the worst-subspace minimax sample threshold over a substantial parameter range, revealing an exponential dependence on degree and precisely quantifying the joint effects of degree, ambient dimension, and subspace dimension. A small modification of our lower-bound construction reveals a strong separation: a single subspace can admit noiseless recovery with optimal sample complexity independent of degree, yet require essentially the worst-case sample size, exponential in degree, to attain the noisy parametric rate. To establish the sampling lower bound, we sharpen the Polyanskiy-Samorodnitsky uncertainty principle (J. Funct. Anal., '19) to uniformly optimal precision and extend its concentration guarantee to large subspaces in which every function concentrates most of its energy on one shared Hamming ball. The sampling upper bound follows from the sharp moment bound of Kirshner and Samorodnitsky (IEEE Trans. Inf. Theory, '21) combined with empirical-process bounds.

翻译：暂无翻译