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 \\(b>0\\) bits per coordinate; we refer to them collectively as EDEN. First, TurboQuant\\(_{text{mse}}\\) is a special case of EDEN obtained by fixing EDEN’s scalar scale parameter to \\(S=1\\). EDEN supports both biased and unbiased quantization, each optimized by a different \\(S\\) (chosen via methods described in the EDEN works). The fixed choice \\(S=1\\) used by TurboQuant is generally suboptimal, although the optimal \\(S\\) for biased EDEN converges to $1$ as the dimension grows; accordingly TurboQuant\\(_{text{mse}}\\) approaches EDEN’s behavior for large \\(d\\). Second, TurboQuant\\(_{text{prod}}\\) combines a biased \\((b-1)\\)-bit EDEN step with an unbiased 1-bit QJL quantization of the residual. It is suboptimal in three ways: (1) its \\((b-1)\\)-bit step uses the suboptimal \\(S=1\\); (2) its 1-bit unbiased residual quantization has worse MSE than (unbiased) 1-bit EDEN; (3) chaining a biased \\((b-1)\\)-bit step with a 1-bit unbiased residual step is inferior to unbiasedly quantizing the input directly with \\(b\\)-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 \\(S\\)) is more accurate than TurboQuant\\(_{text{mse}}\\), and unbiased EDEN is markedly more accurate than TurboQuant\\(_{text{prod}}\\), often by more than a bit (e.g., 2-bit EDEN beats 3-bit TurboQuant\\(_{text{prod}}\\)). We also repeat all accuracy experiments from the TurboQuant paper, showing that EDEN outperforms it in every setup we have tried.<\/p>\n","protected":false},"excerpt":{"rendered":"
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