https://arxiv.org/abs/2604.18555

"This note clarifies the relationship between the recent TurboQuant work and the earlier DRIVE (NeurIPS 2021) and EDEN (ICML 2022) schemes. DRIVE is a 1-bit quantizer that EDEN extended to any bits per coordinate; we refer to them collectively as EDEN. First, TurboQuant is a special case of EDEN obtained by fixing EDEN's scalar scale parameter to . EDEN supports both biased and unbiased quantization, each optimized by a different (chosen via methods described in the EDEN works). The fixed choice used by TurboQuant is generally suboptimal, although the optimal for biased EDEN converges to as the dimension grows; accordingly TurboQuant approaches EDEN's behavior for large . Second, TurboQuant combines a biased -bit EDEN step with an unbiased 1-bit QJL quantization of the residual. It is suboptimal in three ways: (1) its -bit step uses the suboptimal ; (2) its 1-bit unbiased residual quantization has worse MSE than (unbiased) 1-bit EDEN; (3) chaining a biased -bit step with a 1-bit unbiased residual step is inferior to unbiasedly quantizing the input directly with -bit EDEN. Third, some of the analysis in the TurboQuant work mirrors that of the EDEN works: both exploit the connection between random rotations and the shifted Beta distribution, use the Lloyd-Max algorithm, and note that Randomized Hadamard Transforms can replace uniform random rotations. Experiments support these claims: biased EDEN (with optimized ) is more accurate than TurboQuant, and unbiased EDEN is markedly more accurate than TurboQuant, often by more than a bit (e.g., 2-bit EDEN beats 3-bit TurboQuant). We also repeat all accuracy experiments from the TurboQuant paper, showing that EDEN outperforms it in every setup we have tried."