Nearly-linear row count for the canonical SRHT

Establish a row count of O(\epsilon^{-2}d\operatorname{polylog}(n)) for the canonical one-shot subsampled randomized Hadamard transform while retaining the fixed-direction coordinate-wise regression guarantee, without using the padded block construction of the balanced Gaussian-pooled transform.

Background

The paper presents the rigorous dependency-safe baseline for the canonical one-shot SRHT due to Price, Song, and Woodruff, which achieves the fixed-direction regression conclusion with a row count of d^{1+o(1)} rather than d\operatorname{polylog}(n). The paper's balanced Gaussian-pooled transform attains a nearly linear row count by introducing a padded internal dimension, randomized permutation, and disjoint Gaussian pooling.

The unresolved problem is to obtain the desired d\operatorname{polylog}(n) row count directly for the canonical SRHT, without relying on the padded block construction. The authors indicate that solving it would require a new argument, potentially involving an anisotropic inverse-Gram estimate or fluctuation-averaging estimate.