Identify geometric predictors of a subspace’s sampling threshold

Identify geometric properties of a given known subspace V of degree-at-most-k Boolean functions that predict its sample threshold before sampling, distinguishing subspaces that require exponentially many samples in k from subspaces such as spans of Walsh characters that attain the parametric rate after O(m(\log m+t)) samples.

Background

The paper analyzes the worst admissible subspace and shows that some low-degree subspaces can concentrate many directions on a rare Hamming ball, causing an exponential sampling cost under noise. In contrast, the appendix shows that spans of Walsh characters have uniformly bounded feature norms and achieve stable sampling and parametric prediction with roughly m(\log m+t) samples.

The unresolved question is to formulate intrinsic geometric criteria for an individual supplied subspace V—rather than worst-case criteria over all subspaces—that allow its sampling threshold to be predicted before observations are collected.