SplitQuant: Layer Splitting for Low-Bit Neural Network Quantization

Fangzhen Lin; Jaewoo Song

REVIEW 3 major objections 6 minor 18 references

SplitQuant: Layer Splitting for Low-Bit Neural Network Quantization

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read SplitQuant rewrites each quantizable layer as three mathematically equivalent narrower layers, so low-bit quantizers keep outlier signals while gaining finer resolution; on two BERT-Tiny models, INT2 accuracy rises to within about half a…

desk verdict A clean, honest, small idea — k-means layer splitting for better INT2 resolution — with a sound equivalence argument but thin evidence: one good held-out test and one train-set evaluation that should not be in the abstract. read the letter →

arxiv 2501.12428 v2 pith:QP7OHJYJ submitted 2025-01-21 cs.LG cs.AI

Jaewoo Song , Fangzhen Lin This is my paper · ORCID

classification cs.LGcs.AI

keywords quantizationlow-bitoutlierlayersplittingk-meansclusteringlanguagemodeltinyMLEdgeAI

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Deep-learning quantization has to trade off two things: clipping outliers keeps the value range small enough for fine resolution, but throws away strong signals. SplitQuant is the paper's way around that trade: it rewrites each quantizable layer as three mathematically equivalent layers whose individual value ranges are narrower than the original, and lets the quantizer scale each split layer separately. In the reported experiments, this preprocessing lifts INT2 accuracy on two fine-tuned BERT-Tiny text models from 86.5% to 89.8% and from 96.2% to 98.3%, nearly matching the original FP32 models' 90.2% and 98.4%. The paper positions SplitQuant as a model-reshaping preprocessing step that any quantization algorithm can run before its own quantization.

What carries the argument

The central object is the three-way layer split: replace $Wx+b$ by $\sum_{i=1}^3 (W_i x + b_i)$ with $W_i,b_i$ zero-padded cluster masks, and replace an activation layer of length $n$ by three activation layers of length $n/3$ followed by concatenation. The mechanism that carries the argument is the quantization scale identity $S=(2^b-1)/(\alpha-\beta)$: reducing a layer's min-max range by splitting increases $S$, so the same integer grid represents distinct floating-point values more finely, and outliers stay in the model instead of being clipped. K-means clustering, initialized with k-means++, chooses the weight and bias split so that the three ranges are separated; activations cannot be clustered, so they are split by position.

What would settle it

Run the INT2 pipeline on the same models while forcing all three split layers to share one scale and zero-point, or inspect the quantizer's recorded per-layer scale factors: if the three scales are equal, the mechanism is not active and the reported gains should disappear.

Watch

Extended reading notes

Core claim

On its own terms, SplitQuant's central discovery is that outlier clipping is not the only way to improve quantization resolution: the model itself can be reshaped so that each piece of a layer has a narrower range. A linear or convolution layer with weight $W$ and bias $b$ is replaced by three layers whose weights and biases come from k-means clusters of the original parameters, with zeros filling the positions belonging to other clusters, so that the sum of the three layers equals the original $Wx+b$. Activation layers are split into three consecutive chunks whose outputs are concatenated. Each split layer has a smaller $\alpha-\beta$ range, which raises the quantization scale factor $S=(2^b-1)/(\alpha-\beta)$ and therefore the resolution, while no outlier is discarded. The paper reports INT2 accuracy rising by 3.3 and 2.1 percentage points on the two models, with the gains shrinking to near zero at INT8.

Load-bearing premise

SplitQuant only yields finer resolution if the quantization tool assigns an independent scale and zero-point to each of the three split layers; if it computes one range over the original layer or shares a scale across the split, the narrow ranges do not turn into finer resolution.

Editorial extensions

If this is right

For INT2 on the emotion model, accuracy goes from 86.5% to 89.8%, within 0.4 points of the 90.2% FP32 baseline.
For INT2 on the spam model, accuracy goes from 96.2% to 98.3%, within 0.1 points of the 98.4% FP32 baseline.
The gain tracks resolution: INT4 improves by 0.2 and 0.1 percentage points and INT8 by 0.1 and 0.0, so the benefit grows as bit-width shrinks.
SplitQuant is designed as a preprocessing step, so an existing quantization algorithm can be applied unchanged after the split.
Model size can grow up to three times because the three layers replace one, but the added weights are zeros, leaving room for sparse inference to recover the cost.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same per-layer-scale mechanism should transfer to any quantization backend that quantizes each layer independently, so SplitQuant is likely to help other low-bit methods and architectures beyond the two text models tested here; the paper does not run those experiments.
Because the activation split is fixed at one-third chunks, an adaptive split that separates extreme activation values into different chunks could produce larger gains; this variant is testable and not explored in the paper.
If sparse-inference engines exploit the zero-padded split matrices, the threefold layer expansion may become nearly free, which would make the method more practical on edge devices; the paper mentions this possibility but does not implement it.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: SplitQuant is a small paper with a real, honestly-presented idea. You k-means cluster a linear or conv layer's weights and biases into lower/middle/upper groups, split the layer into three zero-injected sublayers, and partition activation layers into thirds. The three layers are exactly equivalent to the original before quantization, and since each split layer has a narrower value range, its per-layer scale factor is larger and low-bit resolution improves. The equivalence argument is correct; that part checks out. The idea is close to OCS and VS-Quant — the paper says so itself — and the genuinely new twist is the k-means clustering of weights into three clusters. That is a legitimate extension, not a breakthrough.

Credit where it is due: the paper is plainly written, the related work discussion is honest, and it flags its own costs (three times the layers, model size up to 18.75% of FP32, sparse inference engines needed to amortize). There are no fitted constants and no circularity. The emotion-recognition result (INT2 86.5% to 89.8% vs 90.2% FP32, on a real 2000-sample test set) is plausible and the right kind of evidence.

The soft spots are real, though. First, the SMS Spam result is measured on the training set. The paper says outright that the entire UCI dataset was used for fine-tuning and then used to compare accuracies. Its justification — both models were compared in the same environment — conflates a fair method comparison with a valid generalization measurement. That number should not appear in the abstract as evidence of near-FP32 accuracy on unseen data. Remove it and the empirical case rests on one held-out test set, no error bars, no repeated seeds.

Second, the paper assumes the downstream quantizer (Quanto) assigns an independent scale and zero-point to each split layer, and it never verifies that. If Quanto shares a scale across the split layers or quantizes activations globally, the narrower ranges would not produce finer resolution and the gain would not follow. This is checkable in a few lines of code.

Third, minor: there is no comparison against percentile clipping, OCS, or VS-Quant, all cited as related. The paper frames itself as complementary, which softens the criticism, but a baseline would have made the claim believable.

Who benefits: tinyML and edge practitioners quantizing small transformers to INT2. I would bring it to the reading group and I would send it to a serious referee, expecting revision rather than rejection. The math is sound, the honesty is refreshing, and the fixable flaws — redo the spam eval on a proper split, verify the scale behavior, add error bars — are all within reach.

Referee Report

3 major / 6 minor

Summary. SplitQuant is a pre-processing method for low-bit quantization. For each linear or convolution layer, it clusters the weights and biases into three groups with k-means (k=3, k-means++ initialization), creates three zero-masked layers, and sums their outputs; for activation layers, it splits the tensor into three contiguous thirds and concatenates their outputs. The construction is mathematically equivalent to the original layer, and because each split layer has a narrower value range, its quantization scale is larger and resolution finer, while outliers are retained. The paper evaluates the method with Hugging Face Quanto on two fine-tuned BERT-Tiny models (emotion recognition and SMS spam detection) and reports INT2 accuracy improvements of 3.3 and 2.1 percentage points, approaching FP32 accuracy.

Significance. If the reported results are reproducible, SplitQuant is a useful orthogonal contribution: it is a simple, architecture-level way to obtain per-subgroup scaling without custom hardware, it is compatible with existing quantizers, and it does not use test-set-fitted constants (k=3 and the 1/3 activation split are fixed). The mathematical equivalence of the split is the paper's strongest point; the sum/concatenation construction is standard and appears correct. The main weaknesses are empirical: one of the two evaluations is performed on the training set, there are no repeated runs or error bars, and the paper does not verify that the downstream quantizer actually assigns independent scales to the split layers.

major comments (3)

[Section 5, Table 1] The UC Irvine SMS Spam experiment is evaluated on the training data. Section 5 explicitly states that the dataset has no split and that 'the entire dataset used for fine-tuning the model was also utilized to compare the accuracies.' Training-set accuracy is not a valid estimate of generalization, and the argument that both models are compared in the same environment establishes fairness of the comparison but not validity of the accuracy numbers. The 96.2% to 98.3% INT2 result therefore cannot support the abstract's and Section 7's claims of near-FP32 performance on unseen data. Please either construct a proper held-out split (or use k-fold cross-validation) or label the result explicitly as an in-sample diagnostic and remove it from the headline claims.
[Section 4, Figures 2 and 3] The mechanism for the accuracy gain depends on the downstream quantizer assigning an independent scale and zero-point to each of the three split layers. The text says SplitQuant 'applies different scaling factors' and Figures 2/3 illustrate this, but the paper never reports the quantization ranges or scales that Quanto actually computes after the split. If Quanto shares a range across the split layers or quantizes with a global range, the narrower per-layer ranges would not translate into finer resolution and the observed gains would need a different explanation. Please provide the actual per-layer ranges/scales (or confirm from Quanto's code that each split layer is quantized independently).
[Section 5, Table 1] Each accuracy number in Table 1 comes from a single run; no repeated seeds, confidence intervals, or significance tests are reported. The central empirical claim is an improvement of 3.3 and 2.1 percentage points, and the held-out evidence after the SMS Spam correction reduces to one 2,000-sample test set. Please provide repeated runs (with different random seeds) and, ideally, error bars or a paired test over multiple models, at least for the INT2 condition.

minor comments (6)

[Table 1] The table and its caption spell the dataset as 'DIAR.AI' whereas the text uses 'DAIR.AI'; please unify.
[Section 4.2] The activation split assumes that the activation length n is divisible by 3; the paper should state how non-divisible dimensions are handled, since common sizes (e.g., 128 or 512) are not divisible by 3.
[Section 4.2 and Section 5] The paper says that for Quanto, which supports only weight quantization by default, activation layers should not be split, but the experimental section does not state explicitly whether the reported results used weight-only quantization and hence no activation splitting.
[Section 2] The statement that Net2Net 'always increase[s] the number of neurons' is only true for Net2WiderNet; Net2DeeperNet adds layers rather than neurons, so the sentence should be qualified.
[Section 3, Equations (1)-(3)] The paper uses INT() as the rounding function without defining the tie-breaking rule; a one-sentence clarification would improve reproducibility.
[References] The Quanto citation is a documentation URL; a versioned release or code reference would be more stable and useful to readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: SplitQuant's layer-splitting equivalence and resolution gain follow from the quantization formula and unsupervised clustering; no fitted parameter or self-citation bears the central claim.

full rationale

The derivation chain is self-contained and non-circular. SplitQuant's mathematical equivalence is established by construction: weights and biases are clustered into three groups and zero-injected so that the elementwise sum of the three split layers reproduces the original linear or convolution layer exactly in FP32 (Figures 2 and 3), and activation splitting by concatenation preserves the identity function (Figure 1). The claimed resolution improvement is a direct consequence of the paper's own quantization equations, S = (2^b - 1) / (alpha - beta), with the observation that each split layer has a smaller alpha - beta and therefore a larger scaling factor and finer resolution. No parameter is fitted to accuracy targets; k-means clustering is unsupervised and applied only to weights and biases, and the split boundaries are not chosen to maximize test accuracy. There are no load-bearing self-citations: the cited works (Net2Net, OCS, VS-Quant, k-means++, Quanto) are external and used for context or implementation, not to justify the central derivation. The emotion-recognition experiment uses a held-out test set of 2000 samples, providing an external benchmark. The SMS Spam experiment is evaluated on the fine-tuning set, which weakens the generalization claim but is a methodological limitation, not circularity, because no accuracy signal is used to set the method's parameters. The remaining weakness about whether the downstream tool assigns independent scales per split layer is an unverified empirical assumption, not a circular reduction. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim rests on a small number of design choices plus assumptions about how the downstream quantizer treats split layers. No invented entities are introduced.

free parameters (2)

k = 3 for weight and bias clusters = 3
Number of clusters chosen ad hoc; no sensitivity analysis is provided, and it determines how many split layers are created.
activation split ratio = 1/3 = 3 equal parts
Activation layers are split into three equal parts to match the weight split; no sensitivity analysis is provided.

assumptions (3)

domain assumption Quantization of a tensor is modeled as Q(x) = INT(Sx) + Z with a single scale S determined by the layer's min and max (Eq. 1-3).
This is the standard quantization model used throughout the paper; the claimed resolution gain depends on this per-layer affine model.
domain assumption After SplitQuant, quantization algorithms assign independent scale factors to each of the three split layers.
The central benefit in Section 4 depends on each split layer being quantized separately; this is stated but not explicitly verified in the experiments.
domain assumption Outliers in DNN weights, biases, and activations carry important signals worth preserving.
This motivates the design in Section 1; if false, splitting would still reduce ranges but the advantage over clipping would be weaker.

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Cite this review

Pith. "Pith review of SplitQuant: Layer Splitting for Low-Bit Neural Network Quantization." pith.science (2026). https://pith.science/paper/QP7OHJYJ

@misc{pith2026250112428,
  author       = {Pith},
  title        = {Pith review of: SplitQuant: Layer Splitting for Low-Bit Neural Network Quantization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QP7OHJYJ}},
  note         = {Machine review of arXiv:2501.12428}
}

read the original abstract

Quantization for deep neural networks (DNNs) is the process of mapping the parameter values of DNNs from original data types to other data types of lower precision to reduce model sizes and make inference faster. Quantization often maps different original values to a single quantized value because the range of the original values is larger than the range of the quantized values. This leads to the degradation of the accuracy of the quantized DNNs. Outliers are a main cause of the degradation of quantization resolution because they enlarge the range of original values. To solve the problem, the percentile method is often used to clip outliers. However, clipping the outliers has another problem of removing the important and strong signals in the DNNs. This paper proposes SplitQuant to keep the outliers and improve the quantization resolution at the same time. SplitQuant narrows down the range of the original values and mitigates the effect of outliers by splitting each quantizable layer into three mathematically equivalent layers and applies different scaling factors. Especially, weights and biases are clustered into lower, middle and upper clusters for optimized split. By preprocessing DNNs with SplitQuant, quantization algorithms can achieve better results. SplitQuant was applied on two BERT-Tiny models and improved the accuracy of INT2 quantization by 3.3%p and 2.1%p, achieving accuracies comparable to those of the original FP32 models.

Figures

Figures reproduced from arXiv: 2501.12428 by the authors.

**Figure 2.** (A) Original linear layer. (B) SplitQuant runs k [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** (A) Original convolution layer. (B) SplitQuant runs [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 13 canonical work pages

[1]

Tiago A Almeida, José María G Hidalgo, and Akebo Yamakami. 2011. Contribu- tions to the study of SMS spam filtering: new collection and results. InProceedings of the 11th ACM symposium on Document engineering . 259–262

work page 2011
[2]

Tianqi Chen, Ian Goodfellow, and Jonathon Shlens. 2015. Net2net: Accelerating learning via knowledge transfer. arXiv preprint arXiv:1511.05641 (2015)

arXiv 2015
[3]

Steve Dai, Rangha Venkatesan, Mark Ren, Brian Zimmer, William Dally, and Brucek Khailany. 2021. Vs-quant: Per-vector scaled quantization for accurate low-precision neural network inference. Proceedings of Machine Learning and Systems 3 (2021), 873–884

work page 2021
[4]

Jacob Devlin. 2018. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805 (2018)

arXiv 2018
[5]

September 24, 2022

gokuls. September 24, 2022. gokuls/BERT-tiny-emotion-intent/tree/main. https: //huggingface.co/gokuls/BERT-tiny-emotion-intent/tree/main November 28, 2024. 1https://github.com/jaewoosong/splitquant

work page 2022
[6]

Christoph Grunau, Ahmet Alper Özüdoğru, Václav Rozhoň, and Jakub Tětek

work page
[7]

Huggingface. 2024. Huggingface Quanto. https://huggingface.co/docs/ transformers/v4.46.3/en/quantization/quanto November 28, 2024

work page 2024
[8]

February 1, 2021

mrm8488. February 1, 2021. mrm8488/bert-tiny-finetuned-sms-spam-detection. https://huggingface.co/mrm8488/bert-tiny-finetuned-sms-spam-detection No- vember 28, 2024

work page 2021

Show all 18 references

[9]

Hanmin Park and Kiyoung Choi. 2019. Cell division: weight bit-width reduction technique for convolutional neural network hardware accelerators. InProceedings of the 24th Asia and South Pacific Design Automation Conference . 286–291

2019
[10]

Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. 2019. Pytorch: An imperative style, high-performance deep learning library. Advances in neural information processing system...

2019
[11]

Elvis Saravia, Hsien-Chi Toby Liu, Yen-Hao Huang, Junlin Wu, and Yi-Shin Chen
[12]

Iulia Turc, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019. Well-read students learn better: On the importance of pre-training compact models. arXiv preprint arXiv:1908.08962 (2019)

2019 arXiv
[13]

Apache TVM. 2019. Automating Optimization of Quantized Deep Learning Mod- els on CUDA. https://tvm.apache.org/2019/04/30/opt-cuda-quantized November 28, 2024

2019
[14]

Ziheng Wang. 2021. SparseDNN: Fast sparse deep learning inference on CPUs. arXiv preprint arXiv:2101.07948 (2021)

2021 arXiv
[15]

T Wolf. 2019. Huggingface’s transformers: State-of-the-art natural language processing. arXiv preprint arXiv:1910.03771 (2019)

2019 arXiv
[16]

Ritchie Zhao, Yuwei Hu, Jordan Dotzel, Chris De Sa, and Zhiru Zhang. 2019. Improving neural network quantization without retraining using outlier channel splitting. In International conference on machine learning . PMLR, 7543–7552

2019
[2018]

In Proceedings of the 2018 conference on empirical methods in natural language processing

CARER: Contextualized affect representations for emotion recognition. In Proceedings of the 2018 conference on empirical methods in natural language processing. 3687–3697

2018
[2023]

In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

A nearly tight analysis of greedy k-means++. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, 1012–1070

2023

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Tiago A Almeida, José María G Hidalgo, and Akebo Yamakami. 2011. Contribu- tions to the study of SMS spam filtering: new collection and results. InProceedings of the 11th ACM symposium on Document engineering . 259–262

work page 2011

[2] [2]

Tianqi Chen, Ian Goodfellow, and Jonathon Shlens. 2015. Net2net: Accelerating learning via knowledge transfer. arXiv preprint arXiv:1511.05641 (2015)

arXiv 2015

[3] [3]

Steve Dai, Rangha Venkatesan, Mark Ren, Brian Zimmer, William Dally, and Brucek Khailany. 2021. Vs-quant: Per-vector scaled quantization for accurate low-precision neural network inference. Proceedings of Machine Learning and Systems 3 (2021), 873–884

work page 2021

[4] [4]

Jacob Devlin. 2018. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805 (2018)

arXiv 2018

[5] [5]

September 24, 2022

gokuls. September 24, 2022. gokuls/BERT-tiny-emotion-intent/tree/main. https: //huggingface.co/gokuls/BERT-tiny-emotion-intent/tree/main November 28, 2024. 1https://github.com/jaewoosong/splitquant

work page 2022

[6] [6]

Christoph Grunau, Ahmet Alper Özüdoğru, Václav Rozhoň, and Jakub Tětek

work page

[7] [7]

Huggingface. 2024. Huggingface Quanto. https://huggingface.co/docs/ transformers/v4.46.3/en/quantization/quanto November 28, 2024

work page 2024

[8] [8]

February 1, 2021

mrm8488. February 1, 2021. mrm8488/bert-tiny-finetuned-sms-spam-detection. https://huggingface.co/mrm8488/bert-tiny-finetuned-sms-spam-detection No- vember 28, 2024

work page 2021