A Note on TurboQuant and the Earlier DRIVE/EDEN Line of Work
arXiv:2604.18555v1 [cs.LG] 20 Apr 2026
Ran Ben-Basat UCL and Broadcom Michael Mitzenmacher Harvard University
Yaniv Ben-Itzhak Gal Mendelson VMware Research by Broadcom North Carolina State University Amit Portnoy Microsoft
Shay Vargaftik VMware Research by Broadcom
Abstract 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, TurboQuantmse 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 TurboQuantmse approaches EDEN’s behavior for large d. Second, TurboQuantprod 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 TurboQuantmse , and unbiased EDEN is markedly more accurate than TurboQuantprod , often by more than a bit (e.g., 2-bit EDEN beats 3-bit TurboQuantprod ). We also repeat all accuracy experiments from the TurboQuant paper, showing that EDEN outperforms it in every setup we have tried.
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Introduction
On March 24, 2026, Google publicly highlighted TurboQuant [12], their recently accepted ICLR 2026 paper, as a breakthrough in AI memory efficiency in an official blog post [14, 12]. That public framing quickly spilled into financial coverage. Investing.com, carrying Reuters credit, reported on March 25, 2026 that Samsung Electronics fell 4.8% and SK Hynix 5.9%, while U.S.-listed memory peers Micron, SanDisk, Western Digital, and Seagate fell between 3% and 6% [11]. Seoul Economic Daily likewise covered the March 26–27, 2026 selloff and linked it to concerns that lower AI memory requirements could reduce future demand for advanced memory chips, while also reporting the counterargument that cheaper AI may expand overall demand over time [5, 7]. However, as we explain in this note, the TurboQuantmse algorithm is a suboptimal special
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case of the biased variant of the EDEN1 algorithm from ICML 2022, and the unbiased variant of EDEN has better accuracy than that of the unbiased variant TurboQuantprod . We also show that much of the analysis used in TurboQuant previously appeared in the DRIVE [8] and EDEN [9] papers (which we collectively refer to as EDEN) and conduct experiments to empirically compare the algorithms.2 We hope that demonstrating the advantages of using EDEN will support its continued adoption in emerging systems more rapidly.
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Preliminaries
−1 for the quantization and dequantization We use x ∈ Rd for the input vector, and we write √ Q and Q d maps. Following EDEN, we also use ηx := ∥x∥2 , so that ηx R(x) has coordinates on the standardnormal scale after rotation. TurboQuant formulates the reconstruction objective through the meansquared distortion Dmse := E∥x−Q−1 (Q(x))∥22 and the inner-product distortion Dprod := E|⟨y, x⟩− ⟨y, Q−1 (Q(x))⟩|2 in, equations (1)-(2) [12]. EDEN [9] utilizes the vector-normalized mean-squared error vNMSE := E∥x − x̂∥22 /∥x∥22 . These are the same normalization conventions we use throughout this note. The common geometric setup is also the same. After a uniform random rotation, the coordinates become identically distributed. The DRIVE paper [8, Lemma 8, Appendix A.4 of the supplemental material] further explains that the exact distribution of each coordinate with finite d is a shifted Beta distribution that rapidly approaches a normal distribution as d increases [9]. TurboQuant finds the same coordinate distribution in Lemma 1 [12].
Biased and Unbiased Scales in EDEN The key distinction for the present note is the choice of reconstruction scale. EDEN’s unbiased scale is introduced in Theorem 2.1: for Sunb (x, R) =
∥x∥22 , ⟨R(x), Q(ηx R(x))⟩
EDEN proves E[x̂] = x [9]. EDEN then proves a corresponding vNMSE bound in Theorem 2.3 and its asymptotic form in Corollary 2.4. One can alternatively choose the scale factor to minimize the distortion, without the unbiasedness constraint, by setting ⟨R(x), Q(ηx R(x))⟩ Sbias (x, R) = . ∥Q(ηx R(x))∥22 With this setting, x̂ = Sbias (x, R)R−1 Q(ηx R(x)) is the best scalar rescaling of the chosen codeword in squared error. For the one-bit precursor DRIVE, this dichotomy is explicit: Lemma 1 and Theorem 2 analyze the MSE-minimizing scale, while Theorem 3, Theorem 4, and Corollary 1 on analyze the unbiased scale and its distributed-mean-estimation consequences [8]. EDEN generalizes this picture to arbitrary bitwidths: Section 3 chooses the Lloyd–Max quantizer that minimizes scalar MSE, while Section 2.3 and Corollary 2.4 explain how to combine the same scalar quantizer with the unbiased scale [9]. This is exactly the distinction that matters for TurboQuant: TurboQuantmse uses a fixed choice S = 1 that is biased, whereas EDEN shows that an appropriate choice of scale leads to an unbiased result, and further shows how to choose a (different) optimal S for biased results. 1
EDEN was also contributed to Intel’s OpenFL [6, 10] in 2022. We note that the authors of the RaBitQ [4] paper have expressed similar concerns (e.g., [3]) regarding their paper; EDEN and DRIVE also predate the RaBitQ work, which we have recently communicated to the RaBitQ authors. Here, we focus on comparing TurboQuant and EDEN. 2
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TurboQuantmse as EDEN with S = 1
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Our first observation concerns the MSE-oriented TurboQuant construction. Viewed through the EDEN parametrization, TurboQuantmse corresponds to the special case obtained by fixing the EDEN scale parameter to S = 1. To make this relationship explicit, Figure 1 shows a unified pseudocode for TurboQuantmse and (both biased and unbiased) EDEN. Relative to the notation of Section 2, the figure writes the dequantized rotated codeword in the same scale as the rotated vector, namely q := Q(ηx y)/ηx . In this normalization, the common structure is especially transparent: rotate, quantize coordinates, apply the inverse rotation, and reconstruct. The three methods differ only in their choice of the final reconstruction scale S: TurboQuantmse fixes S = 1, EDEN-biased uses the MSE-minimizing scalar, and EDEN-unbiased uses the unbiased scalar from Figure 1 of [9]. The scalar quantizers themselves also coincide. For the one-bit case, the same two-point reconstruction is already explicit in DRIVE: Algorithm 1 reconstructs with values in {±S}, and 1 Lemma 1 gives the biased MSE-minimizing scale explicitly as Sbias = ∥R(x)∥ , so the one-bit DRIVE d reconstruction levels are exactly {±∥R(x)∥1 /d} [8, Lemma 1]. EDEN Section 3, specifically Example 1 and Example 2, gives the standard-normal Lloyd–Max codebooks QI1 ≈ {±0.79788} and QI2 ≈ {±0.45278, ±1.51042} Algorithm √ [9]. TurboQuant (Section 3.1 and p √ 1) lists the same centroids, up√to the paper’s d normalization. For b = 1 it uses ± 2/π/ d, and for b = 2 it uses √ {±0.453/ d, ±1.51/ d} [12]. Further to these similarities, we note that the analysis of EDEN provides tighter bounds. For example, for b = 1, DRIVE shows [8, Theorem 2] that the vNMSE is exactly (1−2/π)(1−1/d), which √ 3π 1 is bounded by 1 − 2/π ≈ 0.363. TurboQuant proved [12, Theorem 1] that Dmse ≤ 2 · 4b , which √ gives 3π/8 ≈ 0.68 for b = 1. The authors mention [12, Theorem 1] that for b = 1, Dmse ≈ 0.36, but this is not proven and seems to be based on empirical observation. As mentioned, these results are derived from the fact that the coordinate distribution after rotation, whose analysis also appears in the DRIVE paper [8, Lemma 8, Appendix A.4 of the supplemental material], follows a shifted Beta distribution (which converges to a normal distribution as the dimension grows large). Correspondingly, once Figure 1 is written in this unified form, the empirical behavior is exactly what one should expect. Fixing S = 1 in TurboQuantmse performs less well than the biased version of EDEN (hereafter referred to as EDEN-biased), which chooses the scale to minimize the resulting distortion. Figure 2 shows this comparison across dimensions and bitwidths. Figure 3 further shows the accuracy gap across bitwidth for a specific d = 128 dimension. For Figures 2, 4, and 5, each plotted point is the mean over paired repetitions using the same lognormal sample seed and the same quantizer seed across the compared methods.3 The displayed √ 95% confidence intervals are 1.96 s/ n, where s is the sample standard deviation of the per-pair metric and n is the number of paired seeds for that dimension. In these sweeps, n = 256 for d ≤ 128, n = 128 for d = 256, n = 64 for d ∈ {512, 1024}, n = 32 for d = 2048, and n = 16 for d = 4096. Across all plotted bitwidths and dimensions, the EDEN-biased curve lies below the TurboQuantmse curve. The gap is most visible in lower dimensions, but it remains throughout the displayed range. In large dimensions, for EDEN-biased S does converge to 1, and correspondingly we see TurboQuantmse does approach EDEN-biased performance for larger dimension. 3
We note that the actual input distribution is irrelevant here as the algorithms randomly rotate the input.
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TurboQuantmse /EDEN pseudocode Setup 1. Generate the shared random rotation matrix Π. 2. Construct the Lloyd–Max centroid codebook c1 , . . . , c2b for the rotated coordinates. Quantize 3. Compute y ← Πx. 4. For each j ∈ [d], set idxj ← arg mink∈[2b ] |yj − ck |. 5. Set qj ← cidxj for each j ∈ [d]. 6. Choose the reconstruction scale S. 7. Send (q, S). Dequantize 8. Regenerate Π from the same seed. 9. Output x̂ ← SΠ⊤ q. Relevant choices of S 10. TurboQuantmse : S ← 1. 11. EDEN-biased: S ← ⟨y, q⟩/∥q∥22 . 12. EDEN-unbiased: S ← ∥x∥22 /⟨y, q⟩.
Figure 1: Unified TurboQuantmse /EDEN pseudocode. This presentation is written in the rotatedcodeword scale q = Q(ηx y)/ηx , so the difference between the three methods is entirely in the reconstruction factor S. Fixing S = 1 recovers TurboQuantmse , while the two choices of S recover EDEN-biased and EDEN-unbiased.
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Unbiased EDEN has better accuracy than TurboQuantprod
We now consider the relationship of TurboQuantprod and EDEN. TurboQuantprod has two logical steps: It first uses TurboQuantmse with b − 1 bits to quantize the input. Then, it calculates the error (the residual vector) and quantizes it with one bit per coordinate using the Quantized Johnson Lindenstrauss (QJL) method [13]. We find the unbiased variation of EDEN (EDEN-unbiased) outperforms TurboQuantprod . Upon examination, we find TurboQuantprod is suboptimal in several ways. 1. Its first stage uses only (b − 1) bits, and that stage is itself just the S = 1 special case of EDEN rather than the EDEN-biased choice that minimizes MSE for the same codeword. 2. Its second stage uses a one-bit QJL quantization of the residual, and this one-bit estimator is provably and empirically much weaker than the earlier one-bit unbiased DRIVE quantizer (vNMSE for large dimension converges to approximately 0.571 for DRIVE vs 1.57 for QJL). 3. In fact, splitting into a biased quantization with b − 1 bits followed by a 1-bit unbiased quantization is less accurate than using all b bits for unbiased quantization with EDEN. Figure 5 compares EDEN-unbiased and TurboQuantprod across dimensions and bitwidths. Throughout the plotted range, TurboQuantprod exhibits substantially larger error than the unbiased EDEN baseline. The separation is large for every shown bitwidth, and it remains large as the dimension increases. Notice that the gap is often worth more than a whole bit (e.g., EDEN with 2 bits is more accurate than TurboQuantprod with 3 bits, and with 3 bits EDEN is better than TurboQuantprod with 4 bits). TurboQuantProd
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We also note that even if one decides to split the quantization to biased b − 1 bits followed by unbiased 1-bit quantization, it is better to use DRIVE than QJL for this purpose. This is visible directly in Figure 4, which isolates the one-bit residual-style estimator. There, the pure QJL estimator used by TurboQuantprod has much larger vNMSE than DRIVE-unbiased (i.e., 1-bit EDEN-unbiased) across the entire displayed range. So even before considering the first-stage split, the one-bit residual mechanism is already a weak choice relative to the earlier one-bit unbiased baseline.
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Reproducing the TurboQuant Paper’s Empirical Behavior
A contribution of the TurboQuant paper is to show that this compression mechanism can be applied in a variety of settings, such as nearest-neighbor queries using a mechanism based on inner products. We have also reproduced the accuracy experiments reported in the TurboQuant paper using their open-source code, while adding EDEN in order to make a comparison.4 Figure 6 shows that the error of EDEN-unbiased is markedly lower for inner product estimation than both TurboQuantmse and TurboQuantprod . It also shows that the MSE of EDEN-biased is comparable to but lower than TurboQuantmse , consistent with our results from Section 3. Figure 7 shows the full distributions of inner-product error. Once again, we observe the same patterns: EDEN-unbiased outperforms TurboQuantprod and EDEN-biased is similar to but better than TurboQuantmse . Inner Product Error
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Figure 6: Aggregate accuracy curves in the reproduced TurboQuant-style presentation. The right panel compares TurboQuantmse with EDEN-biased for MSE, while the left panel compares the inner-product-oriented methods as a function of bitwidth. In both panels, the horizontal axis is the bitwidth b. Two additional reproduced figures reinforce the same conclusion in more specialized settings. Figure 8 shows that the inner-product error of TurboQuantprod remains much more dispersed than that of EDEN-unbiased as the average signal strength changes. Figure 9 shows that the same qualitative gap persists in a downstream nearest-neighbor retrieval task.
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We did not compare the runtimes, but we expect them to be very similar.
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Figure 1 Update: Distribution of inner-product error Bitwidth = 1
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Figure 5 Update: Error variance under changing signal strength Avg IP = 0.01
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Randomized Hadamard Transform
We note that both DRIVE/EDEN and TurboQuant suggest using the Randomized Hadamard Transform (RHT) in practice instead of uniform random rotations, in order to reduce the computational cost of the rotation step. We note that when using RHT, outputs from the RHT step may affect unbiasedness. However, both papers observed that it is essentially as accurate and nearly unbiased in practice. This is not true for adversarial inputs (the DRIVE paper provides an example [8]). We note that our followup work QUIC-FL [1] allows unbiased estimates with a single RHT, and that our recent results [2] indicate that by using two consecutive RHTs, one can get provably nearly-unbiased results, in the sense that the bias vanishes polynomially in the dimension.
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Discussion
The TurboQuant work suggests that compression techniques based on randomized rotations may have important potential applications to improve efficient use of AI memory. This note has shown that some of the compression algorithms and much of the corresponding analysis for the TurboQuant methods appeared previously in the DRIVE and EDEN papers. In fact, the TurboQuantmse algorithm is a special case of the biased variant of the EDEN, and the unbiased variant of EDEN has better accuracy than that of the unbiased variant TurboQuantprod . We hope that demonstrating the advantages of using EDEN, with its scaling factor, will support its continued adoption in emerging systems more rapidly.
References [1] Ran Ben Basat, Shay Vargaftik, Amit Portnoy, Gil Einziger, Yaniv Ben-Itzhak, and Michael Mitzenmacher. Accelerating Federated Learning with Quick Distributed Mean Estimation. In International Conference on Machine Learning, 2024. [2] Ran Ben Basat, William Kuszmaul, Amit Portnoy, and Shay Vargaftik. Preprint in preparation, 2026.
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[3] Jianyang Gao. Turboquant and rabitq: What the public story gets wrong. DEV Community, mar 2026. https://dev.to/gaoj0017/turboquant-and-rabitq-what-the-public-story-gets-wrong1i00. [4] Jianyang Gao and Cheng Long. Rabitq: Quantizing high-dimensional vectors with a theoretical error bound for approximate nearest neighbor search. Proceedings of the ACM on Management of Data, 2(3):1–27, 2024. [5] Seo Jong-gap. Google’s ‘turboquant’ sparks memory stock selloff; industry calls demand concerns overblown. Seoul Economic Daily, 2026. March 26, 2026. Available at article link. [6] OpenFL - Secure Federated AI . eden_pipeline.py source code in OpenFederatedLearning. GitHub repository, 2022. https://github.com/securefederatedai/ openfederatedlearning/blob/develop/openfl/pipelines/eden_pipeline.py. [7] Seoul Economic Daily. Semiconductor stocks plunge on google turboquant — ‘actual effect limited to 2.6x’. Seoul Economic Daily, 2026. March 27, 2026. Available at article link. [8] Shay Vargaftik, Ran Ben-Basat, Amit Portnoy, Gal Mendelson, Yaniv Ben-Itzhak, and Michael Mitzenmacher. Drive: One-bit distributed mean estimation. In Advances in Neural Information Processing Systems 34 (NeurIPS 2021), 2021. [9] Shay Vargaftik, Ran Ben-Basat, Amit Portnoy, Gal Mendelson, Yaniv Ben-Itzhak, and Michael Mitzenmacher. Eden: Communication-efficient and robust distributed mean estimation for federated learning. In Proceedings of the 39th International Conference on Machine Learning (ICML 2022), PMLR 162, 2022. [10] VMware’s Open Source Team. VMware Research Group’s EDEN Becomes Part of OpenFL. VMware Open Source Blog, Nov 2022. https://blogs.vmware.com/opensource/2022/11/ 16/vmware-research-groups-eden-becomes-part-of-openfl/. [11] Ambar Warrick. Samsung, sk hynix slide as google touts ai memory compression tech ‘turboquant’. Investing.com, with Reuters credit, 2026. March 25, 2026. Available at article link. [12] Amir Zandieh, Majid Daliri, Majid Hadian, and Vahab Mirrokni. Turboquant: Online vector quantization with near-optimal distortion rate. In The Fourteenth International Conference on Learning Representations, 2026. https://openreview.net/forum?id=tO3ASKZlok. [13] Amir Zandieh, Majid Daliri, and Insu Han. QJL: 1-bit quantized JL transform for KV cache quantization with zero overhead, 2024. [14] Amir Zandieh and Vahab Mirrokni. Turboquant: Redefining ai efficiency with extreme compression. Google blog, 2026. March 24, 2026.
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