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arxiv: 2605.13768 · v1 · submitted 2026-05-13 · 💻 cs.LG · cs.AI· cs.IT· math.IT

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High-Rate Quantized Matrix Multiplication II

Or Ordentlich , Yury Polyanskiy
Authors on Pith no claims yet

Pith reviewed 2026-05-14 19:26 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.ITmath.IT
keywords quantized matrix multiplicationLLM post-training quantizationwaterfillingweighted mean squared errorhigh-rate distortionGPTQscalar quantizationcovariance matrix
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The pith

WaterSIC uses waterfilling on the input covariance to make scalar quantization of LLM weights basis-independent and within 0.25 bit per entry of the information-theoretic limit.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines quantized matrix multiplication when the covariance matrix of the second factor is known, a setting common in weight-only post-training quantization of large language models. It links the task to weighted mean squared error source coding and shows that waterfilling allocates rates across coordinates to minimize distortion. The WaterSIC scheme, built from scalar integer quantizers, achieves high-rate performance fixed by the determinant of the covariance matrix alone. This makes the distortion immune to random basis rotations and keeps it within a factor of 2πe/12 of the optimal rate-distortion bound. GPTQ with random rotations performs nearly as well on real models such as Llama-3-8B.

Core claim

When the covariance matrix Σ_X is known, waterfilling applied to its eigenvalues produces a quantization scheme whose high-rate distortion depends only on det(Σ_X) and lies within a multiplicative factor of 2πe/12 of the information-theoretic minimum distortion for the weighted mean squared error problem.

What carries the argument

Waterfilling rate allocation on the eigenvalues of Σ_X to set per-coordinate step sizes for scalar integer quantizers.

Load-bearing premise

The high-rate regime approximation accurately describes the distortion achieved at the bit widths used in practical LLM quantization, with the covariance matrix known and stationary.

What would settle it

Direct computation of quantization MSE on Llama-3-8B layer weights at 4 bits per entry, compared against the value predicted from det(Σ_X) scaled by the 2πe/12 factor.

Figures

Figures reproduced from arXiv: 2605.13768 by Or Ordentlich, Yury Polyanskiy.

Figure 1
Figure 1. Figure 1: Illustrating ΣX of activations entering various layers of Llama-3-8B when processing Wikitext-2 dataset. Note that this is an estimate of the rate advantage assumes weight matrices are well modeled by N (0, In). In particular, actual weight matrices were never used for this plot. Note that, while our argument only gives a lower bound on the minimal distortion, it can be shown to be achiev￾able asymptotical… view at source ↗
Figure 2
Figure 2. Figure 2: In generalSIC the n codebooks C1, . . . , Cn ⊂ R whose product is C ⊂ R n can be arbitrary, and each of them is further scaled by the corresponding αi . This scaling can in general be absorbed in the codebooks definition. However, since we allow A to depend on ΣX through the matrix U ∈ R n×n while the codebooks C1, . . . , Cn are not allowed to depend on ΣX, we do not absorb A into the codebooks. In genera… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the generalSIC quantization algorithm. The matrix [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the quantization regions for the [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Performance of several weight-only quantization [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustrating rate advantage of WaterSIC over SIC for [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Illustrating Cholesky diagonals U 2 k,k for a randomly rotated V ⊤ΣXV and accuracy of approximation (33) in terms of spectrum of ΣX. GPTQ with random rotation can be accurately estimated from combining (10) and (33). It is an interesting open problem to estimate worst possible gap (over possible spectra λj ≥ ϵ) between GPTQ with rotation and Wa￾terSIC (which in turn is 0.25-bit away from information￾theore… view at source ↗
read the original abstract

This is the second part of the work investigating quantized matrix multiplication (MatMul). In part I we considered the case of calibration-free quantization, whereas here we discuss the setting where covariance matrix $\Sigma_X$ of the columns of the second factor is available. This setting arises in the ubiquitous task of weight-only post-training quantization of LLMs. Weight-only quantization is related to the problem of weighted mean squared error (WMSE) source coding, whose classical (reverse) waterfilling solution dictates how one should distribute rate between coordinates of the vector. We show how waterfilling can be used to improve practical LLM quantization algorithms (GPTQ), which at present allocate rate equally. A recent scheme (known as ``WaterSIC'') that only uses scalar INT quantizers is analyzed and its high-rate performance is shown to be (a) basis free (i.e., characterized by the determinant of $\Sigma_X$ and, thus, unlike existing schemes, is immune to applying random rotations); and (b) within a multiplicative factor of $\frac{2\pi e}{12}$ (or 0.25 bit/entry) of the information-theoretic distortion limit. GPTQ's performance, in turn, is affected by the choice of basis, but for a random rotation and actual $\Sigma_X$ from Llama-3-8B we find it to be within 0.1 bit (depending on the layer type) of WaterSIC, suggesting that GPTQ with random rotation is also near optimal, at least in the high-rate regime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper is the second part of a study on quantized matrix multiplication, focusing on the setting where the covariance matrix Σ_X of the input columns is known (as arises in weight-only post-training quantization of LLMs). It shows how classical reverse waterfilling can be applied to improve rate allocation in schemes such as GPTQ, and analyzes the WaterSIC scheme (scalar INT quantizers) whose high-rate performance is claimed to be (a) fully characterized by det(Σ_X) and therefore basis-independent, and (b) within a multiplicative factor of 2πe/12 (≈0.25 bit per entry) of the Gaussian rate-distortion bound. Experiments on Llama-3-8B layers indicate that randomly rotated GPTQ lies within 0.1 bit of WaterSIC, suggesting near-optimality in the high-rate regime.

Significance. If the high-rate analysis and finite-rate experiments hold, the work supplies a clean theoretical link between classical reverse-waterfilling source coding and practical LLM quantization. The basis-free characterization of WaterSIC and the explicit 0.25-bit gap to the information-theoretic limit are useful benchmarks; the observation that rotated GPTQ is already close to this limit supplies a concrete, low-overhead route to near-optimal weight-only quantization. The manuscript also demonstrates the value of importing classical rate-distortion tools into the LLM compression literature.

major comments (3)
  1. [High-rate analysis] High-rate analysis section (around the derivation of the 2πe/12 gap): the claim that WaterSIC lies within 0.25 bit/entry of the rate-distortion limit rests on the R→∞ regime in which quantization noise is white, additive, and independent of higher-order moments. At the 3–5 bit per coordinate rates typical after waterfilling for Llama-3-8B layers, overload probability, non-uniform bin loading, and marginal kurtosis can enlarge the gap; the manuscript provides no finite-rate error analysis or simulation that quantifies how much the gap inflates at these rates.
  2. [Experiments] Experimental evaluation (Llama-3-8B results): the comparison between WaterSIC and rotated GPTQ is performed on a single model and a limited set of layers. Because the gap to the information-theoretic bound is asserted to be small (0.1 bit), the result is sensitive to the particular covariance structure of Llama-3-8B; a broader test across multiple model families and bit-widths is needed to support the general claim that rotated GPTQ is near-optimal.
  3. [Basis independence] Section on basis independence: the statement that WaterSIC performance depends only on det(Σ_X) is derived under the high-rate white-noise approximation. It is not immediately clear whether the same invariance holds once finite-rate effects (granularity, overload) are included; a short counter-example or additional derivation at moderate rates would strengthen the claim.
minor comments (2)
  1. [Abstract] The abstract and introduction should explicitly state the bit-width range (e.g., 3–5 bits after waterfilling) for which the 0.25-bit gap is claimed to remain representative.
  2. [WaterSIC description] Notation: the relationship between the waterfilling solution for the weighted MSE problem and the scalar quantizer step sizes used in WaterSIC could be written out more explicitly (one additional displayed equation would suffice).

Simulated Author's Rebuttal

3 responses · 2 unresolved

We thank the referee for the constructive comments, which help clarify the scope and limitations of our high-rate analysis. We address each major point below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [High-rate analysis] High-rate analysis section (around the derivation of the 2πe/12 gap): the claim that WaterSIC lies within 0.25 bit/entry of the rate-distortion limit rests on the R→∞ regime in which quantization noise is white, additive, and independent of higher-order moments. At the 3–5 bit per coordinate rates typical after waterfilling for Llama-3-8B layers, overload probability, non-uniform bin loading, and marginal kurtosis can enlarge the gap; the manuscript provides no finite-rate error analysis or simulation that quantifies how much the gap inflates at these rates.

    Authors: We agree that the 2πe/12 gap is derived under the high-rate white-noise approximation. At the 3-5 bit rates relevant to Llama-3-8B after waterfilling, finite-rate effects such as overload and kurtosis can indeed increase the actual gap. We will revise the manuscript to explicitly state this assumption and its limitations, and we will add a short discussion noting that the experimental results (0.1 bit gap for rotated GPTQ) provide empirical evidence that the inflation remains modest in practice. A full finite-rate analysis is beyond the current scope. revision: partial

  2. Referee: [Experiments] Experimental evaluation (Llama-3-8B results): the comparison between WaterSIC and rotated GPTQ is performed on a single model and a limited set of layers. Because the gap to the information-theoretic bound is asserted to be small (0.1 bit), the result is sensitive to the particular covariance structure of Llama-3-8B; a broader test across multiple model families and bit-widths is needed to support the general claim that rotated GPTQ is near-optimal.

    Authors: The experiments use Llama-3-8B to illustrate behavior on a recent, representative LLM under realistic covariance structures. The claim is presented as suggestive rather than universal. We will add text clarifying the limited scope and that the near-optimality observation holds for this model family in the high-rate regime. Broader validation across additional models is desirable but cannot be completed in the current revision cycle. revision: no

  3. Referee: [Basis independence] Section on basis independence: the statement that WaterSIC performance depends only on det(Σ_X) is derived under the high-rate white-noise approximation. It is not immediately clear whether the same invariance holds once finite-rate effects (granularity, overload) are included; a short counter-example or additional derivation at moderate rates would strengthen the claim.

    Authors: The determinant characterization follows directly from the high-rate analysis where the quantization noise is white and the distortion depends only on the eigenvalues via waterfilling. We will revise the section to note that this invariance is approximate at finite rates and may be perturbed by granularity and overload. A full moderate-rate derivation or counter-example is left for future work, but the high-rate result remains a useful benchmark. revision: partial

standing simulated objections not resolved
  • A complete finite-rate error analysis quantifying gap inflation at 3-5 bits per coordinate
  • Broader experimental validation across multiple model families and bit-widths

Circularity Check

0 steps flagged

No significant circularity; claims rest on classical information-theoretic results

full rationale

The paper applies the standard reverse waterfilling solution from classical rate-distortion theory to allocate rates based on the eigenvalues of Σ_X and invokes the known high-rate scalar quantization gap of 2πe/12 relative to the Gaussian bound. These are independent external results, not derived or redefined inside the paper. The basis-free characterization via det(Σ_X) follows directly from the waterfilling formula without redefinition. GPTQ comparisons rely on external Llama-3-8B weights rather than any fitted parameters or self-citation chains. No load-bearing step reduces by construction to the paper's own inputs or prior self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on the classical reverse waterfilling theorem for weighted MSE source coding and the high-rate quantization approximation; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Reverse waterfilling solution optimally distributes rate for weighted mean squared error source coding
    Invoked to improve rate allocation over uniform allocation in GPTQ.

pith-pipeline@v0.9.0 · 5580 in / 1247 out tokens · 46165 ms · 2026-05-14T19:26:02.994717+00:00 · methodology

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