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arxiv: 2601.17187 · v2 · submitted 2026-01-23 · 💻 cs.IT · cs.AI· math.IT

Recognition: 3 theorem links

· Lean Theorem

High-Rate Quantized Matrix Multiplication I

Or Ordentlich , Yury Polyanskiy
Authors on Pith no claims yet

Pith reviewed 2026-05-16 11:16 UTC · model grok-4.3

classification 💻 cs.IT cs.AImath.IT
keywords quantized matrix multiplicationhigh-rate quantizationrate-distortion tradeoffLLM quantizationweight-activation quantizationinformation theory
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The pith

High-rate quantization theory supplies the exact rate-distortion tradeoff for generic matrix multiplication without calibration data

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

The paper examines quantized matrix multiplication for large language models in the generic setting where both weight and activation matrices must be quantized without any prior statistical information or calibration. It recalls the classical high-rate information-theoretic results that relate quantization rate to average distortion and contrasts these limits with the actual performance of two common practical schemes: absolute-maximum integer quantization and floating-point representation. The authors also derive simple heuristic approximations that accurately predict the distortion achieved by these practical methods. This comparison clarifies the gap between theory and current practice when side information is unavailable.

Core claim

In the generic MatMul setting without a priori statistical information about the factors, high-rate quantization theory gives the fundamental tradeoff between quantization rate and distortion; absmax INT and FP schemes achieve distortions strictly above this curve, with accurate heuristic approximations supplied for their performance.

What carries the argument

High-rate quantization theory, which asymptotically relates quantization rate to mean-squared distortion for high rates in the absence of side information.

If this is right

  • Quantization performance in the generic weight-plus-activation case can be evaluated directly against the high-rate information-theoretic bound.
  • Absmax INT and FP schemes leave a measurable gap to the theoretical optimum that cannot be closed without additional side information.
  • Heuristic approximations for practical schemes enable quick prediction of distortion without full simulation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Closing the observed gap would require either increasing the bit rate or moving to the weight-only setting with second-order statistics as studied in part II.
  • The heuristic formulas suggest floating-point formats may retain an advantage over integer schemes at intermediate rates even without calibration.

Load-bearing premise

The matrix factors behave in a manner that permits the high-rate asymptotic formulas to apply without requiring exact statistical distributions or calibration data.

What would settle it

Measure the empirical mean-squared error for random matrices quantized at rates of 4 to 8 bits per entry using absmax INT, then compare the observed distortion values to the high-rate theoretical bound.

Figures

Figures reproduced from arXiv: 2601.17187 by Or Ordentlich, Yury Polyanskiy.

Figure 1
Figure 1. Figure 1: Distribution of ϵ-quantization errors of a vector N (0, σ2 I10000) with σ 2 = 1/12 and ϵ = 0.1 and effects of shrinkage on MSE. known that the best spherical quantizer is able to achieve approximation guarantee (uniformly over all A) s.t. ∥A − Aˆ∥ 2 F ≤ ∥A∥ 2 F 2 −2R+o(1) = na2 −2R+o(1) . How do we bound worst-case distortion? Suppose that we were lucky and Bˆ = B, then the best bound we can have is from C… view at source ↗
Figure 2
Figure 2. Figure 2: Demonstrating accuracy of FP8 approxima [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Histograms of ∆FP and ∆INT for the setup described in Section III-E. possible to 7 for INT4 or 6 for FP8.7 Consequently, both NVINT4 and NVFP4 actually have rate 4+ 8 16 = 4 1 2 bits per entry. We now make the heuristic argument that the distor￾tion attained by NVFP4 is typically upper bounded by the distortion predicted by the IEIN model for 1 mantissa bit (which has Reff(FPM) = 3.2356). Let us ignore the… view at source ↗
Figure 4
Figure 4. Figure 4: Performance of several quantization schemes with [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: 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 6
Figure 6. Figure 6: 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 6
Figure 6. Figure 6: Illustration of the generalSIC quantization algorithm. The matrix [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of the quantization regions for the [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Performance of several weight-only quantization [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Illustrating rate advantage of WaterSIC over SIC for [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Illustrating Cholesky diagonals U 2 k,k for a randomly rotated V ⊤ΣXV and accuracy of approximation (53) in terms of spectrum of ΣX. with a fully ΣX-oblivious codebook. Note that WaterSIC chooses per-coordinate scales αi ∝ U −1 i,i (Cholesky decomposition of) ΣX and, thus, requires large a ≫ 1 to amortize sending of αi’s in high precision (usually BF16). F. Future of weight-only quantization The high-reso… view at source ↗
read the original abstract

This paper investigates the problem of quantized matrix multiplication (MatMul), which has become crucial for the efficient deployment of large language models (LLMs). We consider a Generic MatMul setting, where both matrices must be quantized (weight+activation quantization) without specific apriori (calibration) statistical information about the factors. We review the fundamental information-theoretic tradeoff between quantization rate and distortion (high-rate theory), and contrast those with the performance of popular quantization schemes (absmax INT and floating-point (FP)), for which we also derive accurate heuristic approximations. Part II of this paper studies the weight-only quantization setup where second-order statistics of the activation matrices are available at the encoder.

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

2 major / 2 minor

Summary. The manuscript reviews the information-theoretic high-rate rate-distortion tradeoff for quantized matrix multiplication in a generic setting without prior statistical information on the factors. It contrasts this theoretical limit with the performance of absmax INT and floating-point quantization schemes, for which heuristic approximations are derived.

Significance. If the heuristic approximations are rigorously justified and accurate, the work provides useful benchmarks for understanding quantization limits in LLMs under weight+activation quantization without calibration data, extending standard high-rate theory to a practical generic MatMul setting.

major comments (2)
  1. [§2] §2 (high-rate theory review): The application of standard high-rate approximations (Bennett integral or 6 dB/bit rule) requires a known source pdf satisfying regularity conditions, yet the generic MatMul setting explicitly disclaims any a priori statistics; the paper must specify the implicit distribution used for the theoretical tradeoff and the INT/FP contrasts.
  2. [§4] §4 (heuristic approximations): The abstract claims 'accurate heuristic approximations' for absmax INT and FP but provides no error bounds, validation against the assumed distribution, or derivation details; this is load-bearing for the central contrast with information-theoretic limits.
minor comments (2)
  1. [Introduction] Introduction: Clarify the distinction between the generic MatMul (no calibration) and the weight-only case deferred to Part II with explicit section references.
  2. [Notation] Notation: Define quantization rate R and mean-squared distortion D with explicit formulas before the high-rate review to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the major comments point by point below, clarifying the assumptions in our generic setting and committing to revisions that strengthen the presentation without altering the core claims.

read point-by-point responses

  1. Referee: [§2] §2 (high-rate theory review): The application of standard high-rate approximations (Bennett integral or 6 dB/bit rule) requires a known source pdf satisfying regularity conditions, yet the generic MatMul setting explicitly disclaims any a priori statistics; the paper must specify the implicit distribution used for the theoretical tradeoff and the INT/FP contrasts.

    Authors: We agree that standard high-rate approximations such as the Bennett integral presuppose a source pdf meeting regularity conditions (e.g., finite moments and smoothness). In the generic MatMul setting we deliberately avoid assuming a specific calibrated distribution to model the no-calibration scenario. The implicit assumption underlying the reviewed theoretical tradeoff is therefore any pdf satisfying those regularity conditions, with the rate-distortion expressions stated in integral form that can be instantiated for particular pdfs when needed. The INT and FP heuristic contrasts are derived from the absmax normalization, which depends only on the realized dynamic range rather than the pdf shape. We will revise §2 to state these conditions explicitly and note that the generic setting uses the general high-rate form without committing to one particular pdf. revision: yes

  2. Referee: [§4] §4 (heuristic approximations): The abstract claims 'accurate heuristic approximations' for absmax INT and FP but provides no error bounds, validation against the assumed distribution, or derivation details; this is load-bearing for the central contrast with information-theoretic limits.

    Authors: We acknowledge that the current manuscript presents the heuristic approximations in §4 without full derivation steps or explicit validation. The approximations follow from modeling the quantization error as additive noise whose variance scales with the squared absmax factor; the resulting closed-form distortion expressions are obtained by direct moment calculations under the normalization. We will add a detailed derivation appendix and include Monte-Carlo validation plots comparing the heuristics to exact distortion for representative entry distributions. Analytical error bounds that remain valid for arbitrary distributions are difficult to obtain without further assumptions, so we will instead strengthen the empirical evidence for accuracy in the revised version. revision: partial

Circularity Check

0 steps flagged

No significant circularity: derivation draws from standard high-rate information theory

full rationale

The paper reviews the established information-theoretic high-rate quantization tradeoff (standard Bennett integral and 6 dB/bit scaling) and contrasts it with absmax INT and FP schemes by deriving heuristic approximations. These steps rely on classical rate-distortion results external to the present work rather than any self-definition, fitted-parameter renaming, or load-bearing self-citation chain. The generic MatMul setting is analyzed under the high-rate regime without introducing equations that reduce to the paper's own inputs by construction. No quoted derivation step equates a claimed prediction to a fitted quantity or imports uniqueness solely from the authors' prior unverified results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of classical high-rate quantization theory to matrix multiplication distortion; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption High-rate quantization theory governs the rate-distortion behavior of matrix multiplication under generic (no calibration) conditions
    The paper explicitly reviews this tradeoff as the foundation for its comparisons and approximations.

pith-pipeline@v0.9.0 · 5403 in / 1222 out tokens · 53523 ms · 2026-05-16T11:16:04.954721+00:00 · methodology

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Reference graph

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