The Lattice Geometry of Neural Network Quantization: A Short Equivalence Proof of GPTQ and Babai's Algorithm
Authors: Johann Birnick
Venue: ICLR 2026 Poster (2026)
Keywords: quantization
lattices
GPTQ
Babai
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: We prove that the GPTQ quantization algorithm is equivalent to Babai's nearest-plane algorithm.

Abstract: We explain how data-driven quantization of a linear unit in a neural network corresponds to solving the closest vector problem for a certain lattice generated by input data. We prove that the GPTQ algorithm is equivalent to Babai's well-known nearest-plane algorithm. We furthermore provide geometric intuition for both algorithms. Lastly, we note the consequences of these results, in particular hinting at the possibility of using lattice basis reduction for improved quantization.

Published as a conference paper at ICLR 2026
THE LATTICE GEOMETRY OF NEURAL NETWORK QUANTIZATION: A SHORT EQUIVALENCE PROOF OF GPTQ AND BABAI'S ALGORITHM
Johann Birnick
Department of Mathematics University of California San Diego San Diego, CA 92093, USA
jbirnickeucsd.
ABSTRACT
We explain how data-driven quantization of a linear unit in a neural network cor- responds to solving the closest vector problem for certain lattice generated by input data. We prove that the GPTQ algorithm (Frantar et al. 2023) is equivalent to Babai's well-known nearest- pearest-plane algorithm (Babai 1986). We furthermore
provide geometric intuition for both algorithms. Lastly, we note the consequences of these results, in particular hinting the possibility of using lattice basis reduc- tion for improved quantization.
QUANTIZATION AND LATTICES
Computations in neural networks are usually carried out in 32-bit or 16-bit floating point arithmetic. In particular, the parameters (weights) of the network are stored in this comparatively high precision.
Quantization is the art of reducing precision, in favor of less memory consumption and faster com- putation, while keeping the accuracy as high as possible. this paper, we are interested only in post-training quantization the weights: We are handed trained neural network, and our goal is to approximate (some of) the parameters of the network with a coarse numerical alphabet, while keeping the accuracy high.
Commonly, this effort focused on the linear parts of the network. That is, we are given a lin- map I Rm represented by weight matrix W E m and seek to find another m x n matrix V, whose entries have lower numerical precision and which "approximates W well". Concretely, this means:
Low numerical precision. We will model V as having integer entries, i.e., V E mxn, Together with simple scaling, this covers the case when the available low-precision alphabet is aZ for some E R. Up to clipping, this models quantization with low-bit integers as a memory/computation unit.
Approximation of W. The approximation problem is carried out in a data-driven context: We are allowed to sample representative inputs 3ı, 22, Tt E R2 of the linear unit W. Indeed, in practice can just take some of the training imputs and send them through the network until they reach the linear unit of interest. We want V to approximate W well on these specific inputs, so we want to minimize:
Σ Way Vayll2
(w., Ve,,
Σ XXW. XV
Here, X E kxx is the matrix with rows E1, Dk, and W.,: is the th row of W. Note that this optimization problem is separable: to minimize the on the right, it suffices to minimize each summand |XWT 1.2 -XV7 separately. This corresponds to quantizing a single neuron at a time.
So the problem

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Problem. Given X
R and w RR, find E 2" so that ||X Xulle 2 is as small as possible.
Lattice view. We will now explain the connection to lattices, A lattice is the Z-span Σb, +...+ Zbn of a set of R-linearly independent vectors bi,... ... b, in R The vectors bi.... bn are called a basis of the lattice. Multiple different bases can produce the same lattice. For example, Z² C R2 is lattice, and two possible bases are {11, 0): (0, })} and {(3, 1), (5,2)) See, for instance, Micciancio & Goldwasser (2002) for more background
lattices.
In the problem above, we can view the columns of X as the basis for a lattice in TR That is assuming these columns are linearly independent; more on that below. Then X can be just viewed in IK* and Xu is a lattice point (an element of the lattice). As v runs through 2", Xe runs through all the lattice points. So the minimization problem above asks to compute a lattice point which is close to Xw. In the lattice community this is known as the (approximate) closest vector problem (CVP).
point
While this is generally NP-hard to solve optimally, decades of research have been devoted to prac- tical algorithms that approximately solve CVP. The common approach is to employ an L.LL-like algorithm for basis reduction (Nguyen & Vallée 2010), followed by Babai's nearest plane algo- rithm (Babai, 1986). We will see in section 2 that GPTQ (Frantar et al. 2023) is exactly equivalent to Babai's 's algorithm (up to possibly reversing the columns of X).
Regularization. Note that the columns of X might not be linearly independent; in particular this will be the case if the number & of calibration inputs is less than the number features n. However, can use the following regularization: Append a scalar multiple of the identity matrix below X, so that
X 11 Iuxn
X'
where μ >> 0
(2)
is used in place of X. The columns of X' are linearly independent, and choosing μ do will lead to the naive quantization v := (w) as the optimal solution.
When μ = vx, this is eguivalent to the >-regularization in Frantar et al. (2023). Indeed, GPTQ works with the matrix X and for regularization it replaces this matrix with X X + AZ instead. But we have:
x"X' = x + (uT) = X
So the X' regularization will yield the same result, but it also admits a lattice interpretation.
In summary, we showed how quantization of linear units reduces to solving the CVP for a lattice generated by input data. One can now apply the full range of CVP algorithms for neural network quantization, provided one can scale them to the large lattices that are involved quantization, see section 3
Concurrent Work. A related paper by Chen et al. (002)) is being published at the same time as this work and shows very similar results to those we present here. We want to emphasize that our work was conducted fully independently and had been in development for an extended period of time. Our proofs follow different approach than Chen et al. (2026) and are shorter; we believe that they offer concise and conceptually elegant perspective.
2 GPTQ IS EQUIVALENT TO BABAI'S ALGORITHM
In this section, we view the GPTQ algorithm (Frantar et al. 2023) and Babai's nearest plane algo- rithm (Babai 1986) as procedures for solving the problem from section I
Problem. Given X Rkxx and w R", find na so that || Xulla 2 is as small as possible.
We will show that the algorithms are equivalent, up to reversing the basis of the lattice. We will first review both GPTQ and Babai's algorithm separately. We will see that they differ in two aspects:
GPTQ works in "parameter space" R., Babai's algorithm works in "data space" T
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As GPTQ progresses to smaller sublattices, it keeps the target contained in the R-span of the sublattice. (In every iteration projects onto this span.) BABAI omits such a projection it's not necessary.
In a nutshell, the two algorithms are related by a composition of linear projections I R" Rn- The core is the map I - RR, which is given by X and projects the "data space" onto the "parameter space". The map I + R== is the projection onto the last n coordinates, in step i of the algorithm, and corresponds to projecting onto the remaining sublattice. Whatever Babai's algorithm does IR* project it down to R and this yields precisely what GPTQ does.
The formal equivalence proof then proceeds by essentially rewriting both algorithms as recursive algorithms and using the insight above.
Notation. We will use the following notation. When A is matrix, then A, denotes its jth column, and A,, denotes its 7th row. With A2j denote the submatrix of A which omits the first columns, and 224,5 denotes the submatrix which omits the first i rows and columns. For vectors a, we denote by a, the 7th coordinate, and aze denotes the subvector which omits the first i i- coordinates.
X and are fixed throughout. We have already seen how one can apply regularization so from now we will assume that X has linearly independent columns.
2.1 GPTQ ALGORITHM REVISITED
The original description of GPTQ FFrnnaar et al. 2023) computes matrix as the Cholesky de- composition (x x)-1:
Li = (x*x)* X)-
(4)
We will now show that this just corresponds to taking a QL-decomposition of X and inverting L. Suppose
X QL
(5)
with Q E _ having orthonormal columns and L E _axx being lower triangular with positive entries on the diagonal. Then we have:
Li ""
(6)
From the uniqueness of the Cholesky decomposition (X* X is positive definite) we get:
T-
So in GPTQ we can also compute İ without Cholesky decomposition; instead we compute the QL-dccomposition of X and then invert L. However, this is mostly useful for theoretical study. In practice, one usually uses lot of calibration data, 1: >> in which case it's more memory-efficient to only accumulate the Gram matrix X X instead of storing the full matrix X.
With this note about L place, the GPTQ algorithm can be described as follows: (see section
1: procedure GPTQ(X, w)
2: Compute QL X.
QL-decomposition with L
3:
i 1+r L
4:
(0) t w
5: for 1,..., ....,, n do
U; f- round(w (i-1))
6:
A + ii w (1-1)
7:
8:
(1) wf 1) +
Tna Lx
end for 10: return e
9:
11: end procedure
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The idea behind this is to find a w' whose first coordinate w1j is fixed to be round((1), and which minimizes || - -Xw"l This can then be applied recursively to the other coordinates. The opti- mization problem to find this w' has the explicit solution
round w1 w1 L1
w
(8)
L11
which then yields the GPTQ procedure described above. Sce/Hassibe al. (9993. Frantar & Alistarh (022), and Frantar et al. (2023) for the history and derivation.
Note that, since Le has the first - coordinates all set to zero, the coordinates of the w(() stabilize i increases. Also, because of the normalization factor A Li, and the definition of A,, the
coordinates that they stabilize to are the coordinates of u. Concretely, we have w!) for i S j. In particular, w()) =
As already noted, one might as well write GPTQ as recursive algorithm:
1: procedure GPTO-REC(X,
2: Compute QL= X.
QL-decomposition with L
3: L --1
4: U1 < round(w1)
5: U22 < GPTQ-REc(X (w 21-m 12)22)
L1,1
6: return v
7: end procedure
Indeed, the equivalence of the two procedures follows from observing that the QI-decomposition of
X>2 is precisely given by Q22 and L22,22, and that the inverse of L22.>2 is given by L22,22
X22 Q22 L22.22
(122,22 ( (L-1)2222 = 122,22
From these equalities one can see that GPTQ-REE equivalent GPTQ. (Formally one could prove it via induction.)
2.2 BABAT'S ALGORITHM
We will now describe Babai's nearest plane algorithm (Babai 1986). which was developed in the context of lattices. Recall that we view the columns of X the basis for n-dimensional lattice in R We want to find a lattice vector close to Xw. The idea is to maintain a target vector t. initialized with t = Xw. Then builds up e by taking inner products of with the Gram-Schmidt basis vectors associated to the lattice basis. This can be interpreted as finding a certain "nearest plane", hence the name; section 2.3 and Nguyen Vallée 2010 Chapter 6).
The normalized Gram-Schmidt basis can seen as the Q-factor a QR-decomposition X. The lengths of the Cram--Shmidtt basis vectors stored the diagonal elements of the R-factor. We will instead use the QL-decomposition here, so that it is compatible with GPTQ; this simply corre- sponds applying the "usual" Babai algorithm (which uses QR-decomposition) on the reversed lattice basis. (See section A details of the relation to the classic algorithm.)
1: procedure BABAI(X, w)
2: Compute QL = X.
QL-decomposition with L,,
3: (10) + Xw
4: for i l,..., n do
U; f round (1(1-1).00) (Q)))
5:
6: {(() + 1(- V;X3
end for 8: return 1
7:
9:
end procedure
BABAI could also written as a recursive algorithm, although then it should not take w as input but rather directly the target vector Xw, which for the recursion would be replaced by t V1 We omit the recursive version of the procedure here; instead will be implicit in the equivalence proof in section 2.4
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Remark. We note that BABAI only depends the intrinsic geometry of the lattice, namely the Gram matrix X" X. therefore invariant under the transformation X + UXX for any U E R'xk for which U acts as the identity the column space of X, as this transformation preserves the Gram matrix X X. Thus, in practice, one would replace X by I,, where Q = X is the QL- decomposition of X. This is much more efficient for memory and computation, because X E R whereas L E R=x*, and k > n, see section B in the appendix. For this paper, however, we chose use X everywhere highlight the distinct spaces which computations are carried out.
2.3 THE UNDERLYING GEOMETRY
We will now provide geometric intuition for what GPTQ and Babai's ' algorithm are doing. Note that there are fundamentally two spaces at play: The "parameter space" IR", in which w (and v) lives, and the "data space" g in which X w (and X') lives. The matrix X can be seen as embedding map R" R which maps the quantization grid Z" to a lattice in R
Parameter Space
Data Space
1
X
R
Figure 1: Two spaces at play. X embeds R into R& mapping the quantization grid 2n to a lattice in RR. GPTQ works in R", on the left. Babai's ss algorithm works in R on the right.
X
Figure 2: GPTQ fixes v1 == round(w)). This restricts to lic the orange plane. It defines a new target weight w/ on the orange line, and then proceeds recursively. The target weight w' not just an orthogonal projection to the orange plane. Instead, the update step implicitly uses the geometry from the lattice; in IR* on the right it indeed corresponds to an orthogonal projec- tion.
Figure Babai's algorithm looks for the near- est plane (parallel to the orange or green plane) identifies the orange plane, and then subtracts appropriate integer multiple of Xi from f, leading to the green point t'. then re- cursively looks for lattice point in the green sublattice which is close to the green point t'.
Figures 2 and 3 demonstrate what happens the first step of each algorithm. As we will prove below, both GPTQ and BABAI compute the same first coordinate of i.c., U1. but they do it in different ways.
Note that BABAI'S new target vector t' does not lie in the R-span of the green sublattice. (In fact, at the very end we will have = Xw - Xx.) One could project it onto that span, and the result for wouldn't t change. Indeed, the difference coming from the projection is, by definition. orthogonal to the green plane, so it doesn't t matter in future computations. We will use this in the equivalence proof below, because GPTQ always does this projection implicitly.
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2.4 PROOF OF EQUIVALENCE
Theorem 2.1. The procedures GPTQ and BABAI are equivalent. That is, for any X E Rkxn (of full column rank) and IR,,, they produce the same output 2".
Proof. We already saw that GPTQ is equivalent to GPTQ-REC. We will now show that both EPTQ-REC and BABAI equivalent to the following procedure, which can be interpreted recursive version of BABAI with an additional projection step to the remaining sublattice as noted in
section 2.
procedure PAOA--PRO--REC(X,
RE w)
2: Compute QL X.
b QL-decomposition with L,, >> 0
3: i I--1
4: tt Xw
5: U1 round( (..11)
L1.1
6: U22 BABAI-PROI-REC( (w = L1)22)
T1.1
7: return v
8: end procedure
GPTQ-REC equivalent to BABAI-PROI-REC, The only difference between the algorithms is how they compute GPTQ just rounds w1, and BABAT-PROI-RES rounds the following quantity:
Q1) (Xw, Qi)Qi)
Q
of Lw
LT.W
w1
(10)
L11 L1,1
L111
I111
L1
BABAI is equivalent to BABAI-PROI-REG This is less obvious. Consider the value of "t" in the first BABAI-PROJ-REC recursion, iee. in the first nested call caused by line 6. It is equal to:
X22 +
W1
L
(11)
L
22
Suppose that this product would equal t X1. Then it would follow by induction that BABAI-
PROJ-REC is equivalent to BABAI.
We will show that the product almost equals v1 Xw X1 up additive term of RQ1, where E some scalar. And indeed this suffices to show the equivalence:
Note that, in future iterations/recussions,, both BABAI and BABAI-PROI- (or "t") to take inner products with Q2, ..., Qn. and these vectors orthogonal to Q1. So one could, for example, modify BABAI so that t()) gets additively shifted by n2Q; for any N₃ R. and the output wouldn't t change. Concretely, let's define:
will only ever use (())
1: procedure BABAI (X, wX)
2: Compute QL = X.
QL-decomposition with L,,
3: (10) Xw
4: for = 1,.... 12 do
5: Vi f- round( (1(1-)) Qu))
Li.s
6: {(0) + (11 U; +K KqQu
7: end for
8: return v
9:
end procedure
The previous paragraph shows that BABAI is equivalent to BABAI,,..... for any Ri, Kn E R. And the equivalence of BABAI-PROI-RE with BABAL,,..... for some specific K Ki,..., Ky E R, which depend on the input (X, w), follows from an induction argument, given the claim above.

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It remains to prove the claim:
X22
w +
11
W1
11
=
X
w +
U1 W1 111
-
X1
w
+
U1
W1
(12)
I1,1
L1,1
L1
23
X
W +
11 W1
L1
81
(13)
11,
11
W1
Xw
X1
+
Q1
(14)
i
=XL1
the previously mentioned
CONSEQUENCES AND FUTURE WORK
The equivalence Babai's algorithm with GPTQ has direct consequences for quantization with GPTQ.
Correct Handling of Quantization Over Multiple Layers. Suppose we have quantized linear unit of neural network, and then we want to quantize another linear unit which comes later in the network. Recall that order to obtain X, we take sample inputs of the neural network, and send them through the network to just before our linear unit. This means the data will pass units that we have already quantized. Usually we want to pass the data through the quantized units to generate the lattice X, while passing through the original units to generate the target Xw. So we now want to find E z which minimizes:
|[ Xool
While for GPTQ it's not obvious how to deal with this modified problem, it's completely obvious for Babai's ' algorithm. Indeed, we just need to set the target vector to Xw. (And work with the lattice generated by X.) If one wants to use GPTQ instead, that corresponds to projecting Xw down onto the R-span of the lattice X; the projected point will be equal to X& for some E R", which should then be used as an input to GPTQ. l.e., one should run GPTQ on vi = X+ X instead of w.
This is what the Qronos (Zhang et al., 2026) algorithm does at its core, and indeed improves the quantization quality, see the experimental results of Zhang et al. (0026).
Theoretical Guarantees. Theoretical guarantees about the output of Babai's algorithm directly carry over GPTQ. First, there is absolute guarantee on the error [[ Xei in terms of the lengths Li of the Gram-Schmidt vectors of the lattice:
Theorem 3.1 (aabai 1986 The outpul of BABAI satisfies ||X Xul1² s E 12
Second, there is relative emor guarantee, relating the error to the minimally achievable error:
Theorem 3.2 (Babai 1986)). The output v of BABAI satisfies [X - -xvll S r min.rez || w- X with
7 1 max ΣDE < Vn 1 max
1
Lij SS L
L3 121
Using Lattice Basis Reduction. Theorem 3.2 suggests that the sequence L1,1, L2.2, ... shouldn't t ever increase much in order to obtain a good result. The classic way to make L1,1- L2,2, not ever increase by much is by performing on the Lg and hence significantly improve the outcome of BABAI/GPTQ in theory (Nguyen & Vallée 2010, Chapter 6, Theorem 3). A straightforward wrapper algorithm looks like this:
LLL-like lattice basis reduction. This would give guarantee
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1: procedure WITHREDUCTIONXX, w)
2: (Xred: T) &- LATTICEBASSSREDCCTONN..
3: [Here T E g is the base change matrix satisfying Xred XT.] Xw)
4: Ured <- (EABAT(Feedl
Abuse notation to pass & instead of w to BABAL.
5:
Tured
6: return v
7:
end procedure
The intuition behind this algorithm simple: What BABAT does is, given lattice basis X and target vector 1, it finds lattice point close to and produces its (integer) coordinates v with respect to the basis X. The "better" the basis X, the closer X will be to Lattice basis reduction provides "good" basis Xred together with the base change matrix T E zax" satisfying Xred XT. Then we use BABAI on the reduced basis Xee,, but with the same target vector Xw as before, to compute lattice point close to the target, which will be provided coordinates Ured with respect to Xred- Finally, we compute the coordinates of the point with respect to the original basis X by applying T.
We note that if X is not regularized enough, then T' and hence could potentially have large entries. This could be a problem when one needs to clip the values to a small quantization domain. Even without clipping, could result in bad accuracy of the final network, because large entries are symptom of overfitting to the calibration data X.
We leave the experimental evaluation of WITHREDUCTION and related algorithms for future work.
ACKNOWLEDGMENT
We want to thank Rayan Saab for helpful conversations about this research.
REFERENCES
László Babai. On lovász' lattice reduction and the nearest lattice point problem. Combinatorica, (1):1-13, 1986.
Jiale Chen, Yalda Shabanzadeh, Elvir Crnčević, Torsten Hoefler, and Dan Alistarh. The geometry of LLM quantization: GPTQ as babai's 'ss nearest plane algorithm. In The Fourteenth International Conference Learning Representations, 2026. /RUJattps//ppezzereee.... net net/forumm forum?
1d=NFB40GG565
Elias Frantar and Dan Alistarh. Optimal brain compression: framework for accurate post-training quantization and pruning. Advances in Neural Information Processing Systems, 35:4475-4488, 2022.
Elias Frantar, Saleh Ashkboos, Torsten Hoefler, and Dan Alistarh. Optq: Accurate quantization for generative pre-trained transformers. In International Conference on Learning Representationss 2023. http::///aaaasmmatttccccccooooaa http:hhhww...ooroooooooraaa::::.
Babak Hassibi, David G Stork, and Gregory J Wolff. Optimal brain surgeon and general network pruning. In IEEE international conference on neural networks, pp. 293-299, IEEE, 1993.
Daniele Micciancio and Shafi Goldwasser. Complexity of latrice problems: a cryptographic per- spective, volume 1. Springer Science & Business Media, 2002.
Phong Q Nguyen and Brigitte Vallée. The LLL. algorithm. Springer, 2010.
Shihao Zhang, Haoyu Zhang, Ian Colbert, and Rayan Saab. Qronos: Correcting the past by shaping the future... in post-training quantization. In The Fourleenth International Conference on Learn- ing Representations, 2026. URL wttphtttt:/// httptttpst//peeeeeiiiieeee wtwpaarrwwwwe.etttoouuoooeo////e
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A ALGORITHM DESCRIPTIONS
In this section we provide more detail on why our descriptions of BABAI and GPTQ indeed match the original algorithms.
For BABAL, compare with Nguyen & Vallée (2010 Chapter 6, Algorithm 1). To obtain our descrip- tion, needs to apply the following transformations to their algorithm:
Our target (which they call v) is Xw.
They use the notation by to denote the hth Gram-Schmide vector of the lattice basis. This corresponds taking QR-decomposition
QR, and then letting bद R, Then:
(t, b9) (6g b,)
(+)
(1,
(15)
Qi R.. Ra,
So they round the same value as we do.
Take QL-decomposition instead of QR-decomposition and process the loop in reverse order. (This accounts for reversing the basis.)
For GPTQ, take Frantar et al. 2023 Algorithm I) and apply the following transformations:
Remove the regularization factor AI, since assume it's already part of the lattice, as noted just before section 2.1
Note that what they call X is X in our paper.
Instead of computing I-1 as the Cholesky decomposition of (X"x)-1, compute QL- decomposition of X and invert L. This is equivalent, as explained in section 2.]
Choose block size B = co.
B EFFICIENT BABAI ALGORITHM
As explained at the end of section 2.2 in BABAI (and in GPTQ) one can replace X E p by R"xn, where QL X the QL decoppoosiioo of X, in other words, L X X the reverse Cholesky decomposition of x
Plugging this into the BABAI algorithm yields the following algorithm, which more efficient for memory and computation, because usually k >> n, which helps in lines 3 and 6, and because there no inner product computation in line
1: procedure BABAI- AABAI-CHOLESKY(X w)
2: Compute QL X I = X X. QL- rev. Cholesky decomposition with L4,i
3: {(0) + Lw
4: for = 1,..., n2 do
5: V; t round (1-1) 1..i
6:
{(0) + 1(1-1) biLi
end for 8: return v
9: end procedure


Reviews:
[{"review_id": "F5WgQMwmmj", "reviewer": "Reviewer_8GUU", "summary": "In this work, the authors present a relatively straightforward equivalence proof between GPTQ (a well-known data-driven quantization approach for linear units in a neural network) and the classic Babai’s algorithm (a nearest-plane algorithm). After an in-depth discussion of both algorithms, including visualizations describing the underlying geometry, the equivalence proof is built by constructing recursive versions of both algorithms.\n\nThe paper concludes with a short discussion of the consequences of the equivalence, e.g., easier handling of quantization over multiple layers for GPTQ and the construction of an algorithm using lattice basis reduction.", "questions": "Not at this time.", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper is well written and includes an in-depth discussion of background, making it easy to follow. The presented equivalence has potential for significant impact.", "weaknesses": "While the presented equivalence has potential for significant impact, the paper as it stands does not provide sufficiently strong evidence for this. The final consequences and future work section provides some interesting insights, but to warrant publication at ICLR, I believe that at least some of these observations have to be experimentally validated.\n\nMost important in this regard is showing that the discussed numerical improvements hold and that the WithReduction algorithm provides the improvements the paper hints at. The correct handling of quantization over multiple layers would also be beneficial to study, though it may be hard to demonstrate.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "E5MeUWoxZv", "reviewer": "Reviewer_w5cR", "summary": "This paper establishes a formal connection between the problem of post-training weight quantization for linear layers in neural networks and the classical Closest Vector Problem (CVP) from lattice theory.\n\nThe central contribution of the work is a concise proof demonstrating that GPTQ is mathematically equivalent to Babai's nearest-plane algorithm. The authors discuss several consequences of this equivalence, including potential numerical stability improvements for GPTQ, handling quantization across multiple layers, and leveraging lattice basis reduction techniques to potentially enhance quantization performance.", "questions": "The paper acknowledges \"significant overlap\" with Chen et al. (2025). I do understand Chen et al is a concurrent arXiv paper, and it won't affect the review process of this manuscript. But I'm wondering, could the authors please elaborate on the specific, unique insights that your proof methodology provides?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "I think the paper is easy to follow and has an interesting topic. \n\nIt takes expertise from both domains (Quantization and Lattice Geometry) to notice the equivalence between GPTQ and Babai. And it is valuable that Lattice Geometry knowledge from decades ago can help design better quantization algorithms.\n\nThe proof is more concise than the concurrent work mentioned in this manuscript.", "weaknesses": "The paper is purely theoretical and lacks experimental validation. This is an omission in a field as applied as model compression. The authors propose several compelling consequences in Section 3, most notably the use of lattice basis reduction (LBR) to improve quantization performance. However, these claims remain speculative. Critical questions regarding the computational overhead of LBR, its impact on the final model's generalization (the authors themselves note a risk of overfitting and generating large weight values ), and its actual impact on metrics like perplexity are left unaddressed.\n\nThe paper reads more like a well-written proof-of-concept than a complete ICLR-level publication. While the authors acknowledge an overlap in the manuscript with the concurrent work of Chen et al., it still impacts the novelty since the major conclusions of the two papers are the same.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GYWX6TveFb", "reviewer": "Reviewer_K3b5", "summary": "NA", "questions": "NA", "limitations": "", "rating": null, "confidence": 1, "soundness": 1, "presentation": 1, "contribution": 1, "strengths": "NA", "weaknesses": "NA", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "KIJLEYnG21", "reviewer": "Reviewer_Xcva", "summary": "The authors show that a recently proposed algorithm for neural network quantization, GPTQ, is equivalent to Babai's well-established nearest-plane algorithm appearing in lattice problems. \nBased on this equivalence and the long history of Babai's algorithm, they highlight that the algorithm is associated to known guarantees, and they describe possible improvements and extensions including a) the possibility of a pre-processing step involving basis reduction with the LLL algorithm, to improve performance guarantees; b) the extension to multilayer quantization.", "questions": "Can you detail why the algorithms you detail exactly match GPTQ and Babai's algorithm ?\n\nMinor suggestions: \n-elaborate on the impact of your viewpoint on the analysis of Qronos, which seems to be a distinctive feature of your analysis compared to that of Chen et al.\n-include an explicit summary of the numerical improvements \"mentioned in the paper\" in section 3.\n-write more explicitly that \"what `t' is in the first recursion\" (page 6) means \"in the first call to line 6 of BABAI-PROJ-REC\"\n-clarify in section 2.3 what the \"green sub lattice\" is.\n-to help the reader it may be useful to state after (3) that from now on k always assumed to be larger than n.\n-top of page 3: the \"projection\" is a linear map, not a projection, right ?\n-below (10): writing explicitly X_{>=2} (w+...) as an equation would seem clearer than spelling it out with words, and would ease the connection with (11); \n-spell out GPTQ the first time this acronym is used", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "Showing that a recent empirical approach to neural network quantization is in fact equivalent to a well-established tool from lattice geometry is important. Besides bringing clarification to the field, this also establishes connections that are potentially very fruitful: the paper notably mentions existing guarantees, and describes a number of convincing potential improvements. Some of these consequences also involve the analysis of Qronos, another recent quantization approach.  The paper is well-written, concise and clear.  The proof technique (proving equivalence to Babai-Proj-Rec, and to Babai_{\\kappa_i} is very neat.", "weaknesses": "There is overlap between the contribution of this paper and a recent preprint by Chen et al. Chen et al provide numerical experiments, this paper does not.  Yet, the authors are explicit about this, both papers seem to have been produced at about the same time, and the absence of experiments in this submission actually also contributes to its elegance, concision, and focused viewpoint, which are all rather strengths. \n\nIt is not completely straightforward to check that the paper's description of GPTQ and Babai's algorithm fully matches the description of these algorithms in the corresponding references (by the way a chapter number would have helped when referring to Nguyen and Vallée).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)