CAT-Q: Cost-efficient and Accurate Ternary Quantization for LLMs

# CAT-Q: Cost-efficient and Accurate Ternary Quantization for LLMs
Source: [https://arxiv.org/html/2606.26650](https://arxiv.org/html/2606.26650)
###### Abstract

In this paper, we present CAT\-Q,Cost\-efficient andAccurateTernaryQuantization, for compressing and accelerating LLMs\. Unlike existing state\-of\-the\-art ternary quantization methods that rely on data\-intensive and costly quantization\-aware training to mitigate severe performance degradation, CAT\-Q is a simple yet effective post\-training quantization scheme that is readily applicable to LLMs with diverse architectures and model sizes\. It has two key components, learnable modulation \(LM\) and softened ternarization \(ST\), which are coupled from an optimization perspective\. LM leverages a composition of learnable factors to modulate the distribution of pre\-trained high\-precision weights and the ternary threshold, making them less sensitive to ternarization\. ST further introduces a differentiable transition function to guide the ternarization process toward stable convergence\. We show that, for pre\-trained LLMs with 1\.7B to 8B parameters, CAT\-Q can efficiently quantize them into ternary models using only 512 calibration samples, while achieving superior performance than the seminal BitNet 1\.58\-bit v1 and v2 families \(with 1\.3B to 7B parameters\) trained with 100B tokens, yielding about a 100,000×\\timesreduction in training tokens\. Moreover, we show for the first time that CAT\-Q can quantize much larger pre\-trained LLMs having 14B to 235B parameters into leading ternary models within just 8 to 60 hours on 8 A100\-80GB GPUs\. Code is available at https://github\.com/IntelChina\-AI/BitTern\.

Machine Learning, ICML

## 1Introduction

Large language models \(LLMs\)\(Vaswaniet al\.,[2017](https://arxiv.org/html/2606.26650#bib.bib8); Brownet al\.,[2020](https://arxiv.org/html/2606.26650#bib.bib14); Achiamet al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib15); Anilet al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib13); Liuet al\.,[2024a](https://arxiv.org/html/2606.26650#bib.bib10); Hurstet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib100); Jaechet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib19); Yanget al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib21); Guoet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib11); Yanget al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib83); Singhet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib102); Comaniciet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib99)\)have demonstrated remarkable performance across a wide range of language modeling and reasoning tasks\. However, their large sizes incur significant memory and computational costs, posing a major obstacle to deploy them in real\-world applications, especially on resource\-constrained devices\. Various techniques\(Zafriret al\.,[2019](https://arxiv.org/html/2606.26650#bib.bib69); Maet al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib97); Sunet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib98); Sanhet al\.,[2019](https://arxiv.org/html/2606.26650#bib.bib92); Hsiehet al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib93); Zhanget al\.,[2021](https://arxiv.org/html/2606.26650#bib.bib96); Hsuet al\.,[2022](https://arxiv.org/html/2606.26650#bib.bib95)\)have been proposed to reduce model size and accelerate inference, among which quantization is particularly appealing due to its effectiveness and ease of implementation\.

The ultimate optimization objective of quantization is to reduce the numerical precision of model parameters while preserving model performance, either via post\-training quantization \(PTQ\) or quantization\-aware training \(QAT\)\. Most existing methods\(Shenet al\.,[2020](https://arxiv.org/html/2606.26650#bib.bib41); Dettmerset al\.,[2022](https://arxiv.org/html/2606.26650#bib.bib39); Yaoet al\.,[2022](https://arxiv.org/html/2606.26650#bib.bib62); Frantaret al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib46); Xiaoet al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib59); Shaoet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib4); Linet al\.,[2024b](https://arxiv.org/html/2606.26650#bib.bib42); Cheeet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib50); Ashkbooset al\.,[2024b](https://arxiv.org/html/2606.26650#bib.bib5); Sunet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib61); Wanget al\.,[2026](https://arxiv.org/html/2606.26650#bib.bib119)\)rely on PTQ, which quantizes pre\-trained LLMs using a small set of calibration samples and does not require the highly expensive training or re\-training pipeline used in QAT, and are therefore easy to use in practice\. These PTQ methods are generally effective in 8\-bit quantization, and some, such as QuaRot\(Ashkbooset al\.,[2024b](https://arxiv.org/html/2606.26650#bib.bib5)\), FlatQuant\(Sunet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib61)\)and SliderQuant\(Wanget al\.,[2026](https://arxiv.org/html/2606.26650#bib.bib119)\), attain favorable performance even at 4\-bit precision\. However, they still exhibit poor performance in ultra\-low\-bit settings, such as 2\-bit quantization\.

In this paper, we focus on ternary \(also known as 1\.58\-bit\) quantization, an extreme form of quantization, which maps high\-precision weights to\{1,0,−1\}\\\{1,0,\-1\\\}\. It brings more than a 10×\\timesreduction in memory consumption relative to its FP16 counterpart\. Moreover, by introducing sparsity through the zero state, ternary quantization retains the hardware\-friendly arithmetic advantage of binary quantization\(Courbariauxet al\.,[2015](https://arxiv.org/html/2606.26650#bib.bib90); Rastegariet al\.,[2016](https://arxiv.org/html/2606.26650#bib.bib91); Xuet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib68); Huanget al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib36); Shanget al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib37); Liet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib122)\), whereby most expensive floating\-point multiplication operations can be replaced by far more energy\-efficient integer additions and subtractions, while offering richer expressive capacity\. For ternary quantization, it is challenging to devise an effective optimization scheme that can accurately approximate high\-precision weights by only using values of\{1,0,−1\}\\\{1,0,\-1\\\}due to the high risk of information loss\. To alleviate severe accuracy drop in a more straightforward manner, existing state\-of\-the\-art ternary quantization methods, such as BitNet 1\.58\-bit\(Maet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib63); Wanget al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib67)\), TriLM\(Kaushalet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib112)\)and Tequila\(Huanget al\.,[2026](https://arxiv.org/html/2606.26650#bib.bib113)\), unanimously adopt QAT, which enables the model to adapt its parameters to 1\.58\-bit precision by simulating ternarization during full\-precision training\. Despite achieving promising results, they require massive amounts of training tokens \(e\.g\., hundreds of billions\) and entail enormous computational, energy and time costs, limiting their scalability and practicality\. For example, the BitNet 1\.58\-bit v2 family\(Wanget al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib67)\)is trained on 100B tokens, with its largest model containing 7B parameters; while the TriLM family\(Kaushalet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib112)\)is trained on 300B tokens, with a maximum model size of 3\.9B parameters\. Tequila\(Huanget al\.,[2026](https://arxiv.org/html/2606.26650#bib.bib113)\), the most recent ternary quantization method, still requires 10B tokens to quantize a pre\-trained LLM into a ternary model via re\-training\. Furthermore, Tequila is merely evaluated on two small\-scale models: the 1B and 3B instances of the Llama3 family\(Grattafioriet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib18)\)\. In a nutshell, existing top\-performing ternary quantization methods are tailored to a narrow set of LLMs with specific architectures and small model scales, owing to their data\-intensive and costly QAT pipelines\. Although PTQ is preferred in industry as it does not require access to the original training data and eliminates data privacy and security concerns, as well as the need for massive computational resources, there currently exists no PTQ\-based ternarization method that could match the performance of the aforementioned QAT\-based methods\. To bridge this gap, we revisit PTQ for ternary weight quantization from an optimization perspective of studying its major technical challenges\.

Given a small calibration set \(e\.g\., 512 sequences of length 2048 tokens randomly sampled following common practice in the PTQ field, rather than using the original training data\) and a pre\-trained LLM with arbitrary architecture and model scale, we identify that two major challenges are closely related to the severe performance degradation observed when minimizing reconstruction error of ternary weight quantization\. The first challenge is that direct ternarization of pre\-trained high\-precision weights tends to generate a ternary weight distribution that is not well aligned with the high\-precision weight distribution\. This distributional misalignment between the resulting ternary weights and their high\-precision counterparts leads to serious information loss\. Therefore, designing a proper strategy to refine the distributional alignment is critical to reduce reconstruction error\. The second challenge, which is more fundamental, is that ternarization optimization is notably difficult to converge, due to its extremely low\-bit form and non\-differentiable nature\. To address this extreme discretization problem, existing ternary quantization methods for LLMs follow previous strategies\(Liet al\.,[2016](https://arxiv.org/html/2606.26650#bib.bib87); Zhuet al\.,[2017](https://arxiv.org/html/2606.26650#bib.bib88)\)originally designed for ternarizing convolutional neural networks containing significantly fewer parameters\. Specifically, they always use non\-differentiable hard ternarization functions throughout the optimization process, which cannot adapt to complex weight variations and ensure stable convergence, especially in the PTQ regime\.

Motivated by the above analysis, we presentCost\-efficient andAccurateTernaryQuantization \(CAT\-Q\), a new post\-training ternarization method that is applicable to a wide range of pre\-trained LLMs\. CAT\-Q consists of two key components\. The first component, learnable modulation \(LM\), leverages a composition of three learnable factors to refine the distributional statistics of pre\-trained high\-precision weights as well as the ternary threshold based on a small set of calibration samples\. These learnable factors enable an adaptive weight distribution transformation that suppresses the effect of outliers and makes high\-precision weights less sensitive to ternarization, mitigating the distributional misalignment between the resulting ternary weights and their high\-precision counterparts, to a certain extent\. The second component, softened ternarization \(ST\), introduces a novel transition function that guides the ternarization process toward stable convergence via establishing a two\-stage relay of differentiable ternarization and hard ternarization\. Coupling LM and ST into a sliding\-layer quantization pipeline results in a clean and efficient implementation of CAT\-Q\.

We conduct comprehensive experiments to evaluate the efficacy of CAT\-Q across 10 pre\-trained LLMs spanning diverse architectures \(including dense and mixture of experts \(MoE\) models\) and model scales \(1\.7B to 235B parameters\)\(Yanget al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib21),[2025](https://arxiv.org/html/2606.26650#bib.bib83); Touvronet al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib17)\)\. We show that, under similar model scales \(1\.7B to 8B parameters\), ternary weight models generated by CAT\-Q using only 512 calibration samples \(about 1 million tokens\) outperform the seminal BitNet 1\.58\-bit v1 and v2 families trained with 100B tokens, bringing an approximately 100,000×\\timesreduction in training tokens\. Moreover, we show for the first time that CAT\-Q can readily scale to much larger pre\-trained LLMs having 14B to 235B parameters\. When quantizing LLMs with 1\.7B to 235B parameters, CAT\-Q requires only 1 to 60 hours on 8 A100\-80GB GPUs\. In addition, our models with ternary weights and 8\-bit activations get performance comparable to their ternary\-weight\-only counterparts, further demonstrating the broader applicability of our method\.

## 2Method

Figure[1](https://arxiv.org/html/2606.26650#S2.F1)shows an overview of CAT\-Q\. Next, we describe its formulation and key components\.

![Refer to caption](https://arxiv.org/html/2606.26650v1/x1.png)

Figure 1:Overview of the CAT\-Q’s learning flow for ternarizing the weights of a linear layer in any pre\-trained LLM and its hardware\-friendly weight reconstruction for ternary model deployment\. Please see Figure[3](https://arxiv.org/html/2606.26650#S2.F3)for an illustration of the softened ternarization process\.### 2\.1Preliminary Concepts and Motivation

The concept of ternary weight quantization was originally proposed in TWN\(Liet al\.,[2016](https://arxiv.org/html/2606.26650#bib.bib87)\)to train convolutional neural networks from scratch for computer vision tasks\. Concretely, it constrains model weights to\{1,0,−1\}\\\{1,0,\-1\\\}via solving a layer\-wise weight reconstruction problem:

a​r​g​m​i​nα,T​‖W−α​T‖22\.\\underset\{\{\\alpha,\\textbf\{T\}\}\}\{argmin\}\\ \|\|\\textbf\{W\}\-\\alpha\\textbf\{T\}\|\|\_\{2\}^\{2\}\.\(1\)HereWdenotes the high\-precision weights for a linear layer, andα\>0\\alpha\>0denotes a scaling factor to rescale the corresponding ternary weightsTwhose elements are obtained by a hard ternarization function:

Ti=Q​\(Wi;Δ\)=\{1,i​f​​Wi\>Δ0,i​f​​\|Wi\|≤Δ−1,i​f​​Wi<−Δ,T\_\{i\}=Q\(W\_\{i\};\\Delta\)=\\begin\{cases\}1,&if\\text\{ \}W\_\{i\}\>\\Delta\\\\ \\;\\;0,&if\\text\{ \}\|W\_\{i\}\|\\leq\\Delta\\\\ \-1,&if\\text\{ \}W\_\{i\}<\-\\Delta,\\end\{cases\}\(2\)whereWiW\_\{i\}denotes theit​hi^\{th\}element ofWandΔ\>0\\Delta\>0is a threshold\. Subsequent works for ternary LLMs typically follow TWN\. TernaryBERT\(Zhanget al\.,[2020](https://arxiv.org/html/2606.26650#bib.bib103)\)makes an early research effort to ternarize the weights of small BERT\-based language models\(Devlinet al\.,[2019](https://arxiv.org/html/2606.26650#bib.bib9)\)during fine\-tuning, which uses transformer distillation\(Jiaoet al\.,[2020](https://arxiv.org/html/2606.26650#bib.bib94)\)to compensate for severe accuracy degradation\. Instead, existing state\-of\-the\-art ternary LLMs, such as BitNet 1\.58\-bit\(Maet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib63); Wanget al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib67)\), TriLM\(Kaushalet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib112)\)and Tequila\(Huanget al\.,[2026](https://arxiv.org/html/2606.26650#bib.bib113)\), rely on QAT\. In contrast to these works, we focus on ternarizing the weights of LLMs in the more challenging PTQ regime, aiming to strike a substantially better balance between quantization cost and performance, and enabling its broad applicability to LLMs with diverse architectures and model sizes\.

### 2\.2Learnable Modulation

![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b/loss-layer_4-self_attn.v_proj_nolegend.png)Figure 2:Comparison of weight reconstruction errors with the scaling factorα\\alphaand the thresholdΔ\\Deltadetermined by static approximation \(blue dots\), direct learning \(orange dots\) and our learnable modulation \(green dots\)\. Under the same settings, we use the4t​h4^\{th\}layer of Qwen3\-4B for an illustration\.In the Appendix, we provide additional comparisons on different layers across multiple LLMs\.According to the above formulation, the core problem in ternary weight quantization is how to estimate appropriate values for the scaling factorα\\alphaand the thresholdΔ\\Delta\. The pioneering TWN approximates them asα=1\|IΔ\|​∑i∈IΔ\|Wi\|\\alpha=\\frac\{1\}\{\|I\_\{\\Delta\}\|\}\\sum\_\{i\\in I\_\{\\Delta\}\}\|W\_\{i\}\|andΔ=0\.7n​∑i=1n\|Wi\|\\Delta=\\frac\{0\.7\}\{n\}\\sum\_\{i=1\}^\{n\}\|W\_\{i\}\|, whereIΔ=\{i∣1≤i≤n,\|Wi\|\>Δ\}I\_\{\\Delta\}=\\\{\\,i\\mid 1\\leq i\\leq n,\\ \|W\_\{i\}\|\>\\Delta\\,\\\},\|IΔ\|\|I\_\{\\Delta\}\|is the number of elements inIΔI\_\{\\Delta\}, andnnis the number of elements inW\. Recent ternary LLMs, such as BitNet 1\.58\-bit\(Maet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib63); Wanget al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib67)\), TriLM\(Kaushalet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib112)\)and Tequila\(Huanget al\.,[2026](https://arxiv.org/html/2606.26650#bib.bib113)\), simply use the absmean ternarization:α=1n​∑i=1n\|Wi\|\\alpha=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\|W\_\{i\}\|andΔ=α2\\Delta=\\frac\{\\alpha\}\{2\}\. We notice that some prior works\(Rastegariet al\.,[2016](https://arxiv.org/html/2606.26650#bib.bib91); Zhuet al\.,[2017](https://arxiv.org/html/2606.26650#bib.bib88); Chenet al\.,[2024b](https://arxiv.org/html/2606.26650#bib.bib109)\)directly treat the scaling factorα\\alphaand or the thresholdΔ\\Deltaas learnable parameters to train binary or ternary neural networks, while others\(Esseret al\.,[2020](https://arxiv.org/html/2606.26650#bib.bib117); Gonget al\.,[2019](https://arxiv.org/html/2606.26650#bib.bib118); Liuet al\.,[2025b](https://arxiv.org/html/2606.26650#bib.bib111)\)extend such learning paradigm to estimate quantization step sizes under different bit\-width settings\. Inspired by them, we also learnα\\alphaandΔ\\Deltafor ternarizing each weight group of pre\-trained LLMs based on a small number of calibration samples\. However, in the PTQ regime, we empirically find that directly learningα\\alphaandΔ\\Deltastill suffers from the distributional misalignment between the resulting ternary weights and their pre\-trained high\-precision counterparts, showing only modest improvement over static approximation to alleviate severe accuracy degradation

![Refer to caption](https://arxiv.org/html/2606.26650v1/x2.png)

Figure 3:Illustration of the softened ternarization \(ST\) process\. For a linear layer, taking its pre\-trained weightsWas the initialization point \(t=0t=0\), ST employs a learnable two\-stage relay of differentiable ternarization and hard ternarization to ensure stable convergence\. In the first stage, ST produces an asymptotic ternary output by performing continuous quantization based on the transformed weightsW^\\hat\{\\textbf\{W\}\}\. It relies on a novel smooth transition functionf​\(⋅\)f\(\\cdot\)to gradually evolve from the identity mapping to differentiable ternarization via a sequence of continuous mappings with progressively increasing sharpness along the normalized calibration time state0<t≤γ0<t\\leq\\gamma\. In the second stageγ<t≤1\\gamma<t\\leq 1, ST proceeds with hard ternarization to get a final solution\. Notations are clarified in the Method section\.\(see Figure[2](https://arxiv.org/html/2606.26650#S2.F2)for an illustrative comparison\)\. To address this issue, we present learnable modulation \(LM\), the first module of our method, which modulates the pre\-trained weight distribution to be less sensitive to the ternarization via introducing a learnable linear transformation defined as:

W^=W−μα,where​μ=μ0\+δμ​α0,​α=δα​α0\.\\hat\{\\textbf\{W\}\}=\\frac\{\\textbf\{W\}\-\\mu\}\{\\alpha\},\\text\{ where \}\\mu=\\mu\_\{0\}\+\\delta\_\{\\mu\}\\alpha\_\{0\},\\text\{ \}\\alpha=\\delta\_\{\\alpha\}\\alpha\_\{0\}\.\(3\)Here,μ0=1n​∑i=1nWi\\mu\_\{0\}=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}W\_\{i\}denotes the mean ofW,α0=1n​∑i=1n\|Wi−μ0\|\\alpha\_\{0\}=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\|W\_\{i\}\-\\mu\_\{0\}\|is the absolute mean ofW−μ0\\textbf\{W\}\-\\mu\_\{0\}, and−1<δμ<1\-1<\\delta\_\{\\mu\}<1andδα\>0\\delta\_\{\\alpha\}\>0are two learnable factors to refineα0\\alpha\_\{0\}andμ0\\mu\_\{0\}, which also enable the elements of the transformed weightsW^\\hat\{\\textbf\{W\}\}can have reversed signs against the original weightsW\. Besides, for the ternarization function, we haveΔ=δΔ​Δ0\\Delta=\\delta\_\{\\Delta\}\\Delta\_\{0\}, whereδΔ\>0\\delta\_\{\\Delta\}\>0is a learnable factor to adjust the initial thresholdΔ0\\Delta\_\{0\}\. By default,Δ0=0\.5\\Delta\_\{0\}=0\.5as we use the transformed weightsW^\\hat\{\\textbf\{W\}\}\(equivalently,Δ0=α2\\Delta\_\{0\}=\\frac\{\\alpha\}\{2\}corresponding to the pre\-trained high\-precision weightsW\) to determineT\. We introduce a disentangled learning strategy to determine these three factorsδμ\\delta\_\{\\mu\},δα\\delta\_\{\\alpha\}andδΔ\\delta\_\{\\Delta\}\. Specifically, we use the transformed weightsW^\\hat\{\\textbf\{W\}\}as a redistribution proxy to learn the thresholdΔ\\Deltaand the ternary weightsT, while approximating the pre\-trained weights asW≈α​T\\textbf\{W\}\\approx\\alpha\\textbf\{T\}withoutμ\\mu\. This disentangled learning strategy not only preserves the original hardware\-friendly implementation property of TWN but also yields better ternarization performance than the counterpart that approximatesW≈α​T\+μ\\textbf\{W\}\\approx\\alpha\\textbf\{T\}\+\\mu, as validated by an ablation shown in Table[10](https://arxiv.org/html/2606.26650#S3.T10)\. Mathematically, following the form of the ternarization functionQ​\(⋅\)Q\(\\cdot\)defined in Equation[2](https://arxiv.org/html/2606.26650#S2.E2), in the LM component, theit​hi^\{th\}element of the ternary weightsTis obtained by:

Ti=Q​\(W^i;δΔ​Δ0\)\.T\_\{i\}=Q\(\\hat\{W\}\_\{i\};\\delta\_\{\\Delta\}\\Delta\_\{0\}\)\.\(4\)

### 2\.3Softened Ternarization

Due to its extreme low\-bit form and non\-differentiable nature, ternary weight quantization is inherently difficult to optimize and converge\. Following TWN\(Liet al\.,[2016](https://arxiv.org/html/2606.26650#bib.bib87)\), existing QAT\-based ternarization methods for LLMs\(Maet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib63); Wanget al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib67); Kaushalet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib112); Huanget al\.,[2026](https://arxiv.org/html/2606.26650#bib.bib113)\)perform hard ternarization using the straight\-through estimator for gradient approximation, where convergence is largely facilitated by the quantization\-aware training process, particularly for LLMs with small model scales\. However, in the post\-training quantization \(PTQ\) regime, where no constraints are imposed on model architectures or scales, this fundamental challenge becomes significantly more pronounced\. Although, in Figure[2](https://arxiv.org/html/2606.26650#S2.F2), we have shown that learning the scaling factorα\\alphaand the thresholdΔ\\Deltawith the LM component achieves better performance than static approximation and direct learning methods, it still performs hard ternarization \(facing the non\-differentiability issue\) throughout the calibration learning, lacking a technical guarantee for good convergence\. To bridge this gap, we propose softened ternarization \(ST\), the second component of our method inspired by previous works for quantizing convolutional neural networks\(Gonget al\.,[2019](https://arxiv.org/html/2606.26650#bib.bib118); Esseret al\.,[2020](https://arxiv.org/html/2606.26650#bib.bib117)\), which relies on a novel transition function:

f​\(W;s,Δ\)=tanh⁡\(s​\(W−Δ\)\)\+tanh⁡\(s​\(W\+Δ\)\)2​tanh⁡\(s\)\.f\(W;s,\\Delta\)=\\frac\{\\tanh\(s\(W\-\\Delta\)\)\+\\tanh\(s\(W\+\\Delta\)\)\}\{2\\tanh\(s\)\}\.

\(5\)Here,WWdenotes an arbitrary weight input either from the pre\-trained weightsWor its transformed versionW^\\hat\{\\textbf\{W\}\},s\>0s\>0is a coefficient which controls the sharpness of the output curve,Δ\\Deltadetermines the width of its zero\-output region, and the denominator2​tanh⁡\(s\)2\\tanh\(s\)controls the output boundary\. This transition function has several appealing properties: \(1\) for any choice ofss, it has a symmetric output curve about the origin and is differentiable in terms ofWW; \(2\) whens→0s\\to 0, it approximates the identity mappingf​\(⋅\)=Wf\(\\cdot\)=W; \(3\) whenssincreases from a small positive value to a sufficiently large value with a uniform step size, it behaves as a finite sequence of continuous mappings with progressively increasing sharpness, and asymptotically converges toward the ternary output in a differentiable manner\. Empirically, whens=4\.95s=4\.95, the output is already scaled within\[−1,1\]\[\-1,1\]; \(4\) whens→∞s\\to\\infty, it approaches the hard ternarization defined in Equation[2](https://arxiv.org/html/2606.26650#S2.E2)\. We couple this transition function with the LM component, and adapt them across calibration learning iterations, forming our softened ternarization process:

Ti=\{Wi,i​f​t=0f​\(W^i;tγ​s0,δΔ​Δ0\),i​f​0<t≤γQ​\(W^i;δΔ​Δ0\),i​f​γ<t≤1\.\\vskip 0\.0ptT\_\{i\}=\\begin\{cases\}W\_\{i\},&if\\ t=0\\\\ f\(\\hat\{W\}\_\{i\};\\frac\{t\}\{\\gamma\}s\_\{0\},\\delta\_\{\\Delta\}\\Delta\_\{0\}\),&if\\ 0<t\\leq\\gamma\\\\ Q\(\\hat\{W\}\_\{i\};\\delta\_\{\\Delta\}\\Delta\_\{0\}\),&if\\ \\gamma<t\\leq 1\.\\end\{cases\}\\vskip 0\.0pt\(6\)Letmmdenote the total number of calibration epochs \(by default, we setm=60m=60when using 512 calibration samples\) for ternarizing the weights of a pre\-trained LLM\. Here,γ\\gammadenotes the calibration epoch ratio allocated for differentiable ternarization based on a finite sequence of continuous mappings,1−γ1\-\\gammadenotes the calibration epoch ratio allocated for hard ternarization,ttdenotes the normalized time state of the current calibration epoch, ands0\>0s\_\{0\}\>0is a constant to initialize the sharpness \(by default, we sets0=30s\_\{0\}=30\) of the transition functionf​\(⋅\)f\(\\cdot\)\. Specifically, for a linear layer, taking its pre\-trained high\-precision weightsWas the starting point, the ST process undergoes two stages\. In the first stage, ST produces an asymptotic ternary output by performing continuous quantization from the identity mapping to differentiable ternarization of the transformed weightsW^\\hat\{\\textbf\{W\}\}via a finite sequence of continuous mappings, which is governed by the transition functionf​\(⋅\)f\(\\cdot\)with progressively increasing sharpness along the normalized calibration time state0<t≤γ0<t\\leq\\gamma\. Intriguingly, the asymptotic ternary output from the first stage closely approximates a hard ternarization solution\. Naturally, in the second stage, ST proceeds with hard ternarization to get a final solution, taking the gradients computed in the last iteration of the first stage for subsequent updates of three learnable factors\. By this two\-stage relay of differentiable ternarization and hard ternarization, ST establishes a smooth transition that effectively guides the ternarization process toward stable convergence\. Figure[3](https://arxiv.org/html/2606.26650#S2.F3)shows an illustration of the ST process\.

### 2\.4Sliding\-Layer Ternarization Optimization

In implementation, we adopt a sliding\-layer output reconstruction scheme instead of the predominant layer\-wise weight reconstruction used in existing LLM ternarization methods\. We are inspired by recent works\(Liuet al\.,[2024b](https://arxiv.org/html/2606.26650#bib.bib56); Dinget al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib54); Wanget al\.,[2026](https://arxiv.org/html/2606.26650#bib.bib119)\)that show quantizing multiple layers together tends to yield reduced quantization errors against quantizing a single layer as it makes neighboring layers be aware of each other, enabling to use layer dependencies to mitigate information loss\. However, to the best of our knowledge, this methodology has not yet been explored for ternarizing LLMs\. Motivated by this, we combine CAT\-Q with the framework of SliderQuant\(Wanget al\.,[2026](https://arxiv.org/html/2606.26650#bib.bib119)\)to form our output reconstruction objective for ternarizing the weights of LLMs in the PTQ regime\.

Let𝒲=\{W1,…​Wl\}\\mathcal\{W\}=\\\{\\textbf\{W\}\_\{1\},\.\.\.\\textbf\{W\}\_\{l\}\\\}denote the set of high\-precision weights for the current sliding window consisting oflllayers in a pre\-trained LLM, and letXdenote its input feature corresponding to a small set of calibration samples\. Then, the optimization objective of our CAT\-Q is to minimize anL2L\_\{2\}\-normed loss function defined as:

a​r​g​m​i​n𝒜,𝒯​‖ℱ​\(𝒲,X\)−ℱ​\(𝒜⋅𝒯,X\)‖22\.\\underset\{\\mathcal\{A\},\\mathcal\{T\}\}\{argmin\}\\ \|\|\\mathcal\{F\}\(\\mathcal\{W\},\\textbf\{X\}\)\-\\mathcal\{F\}\(\\mathcal\{A\}\\cdot\\mathcal\{T\},\\textbf\{X\}\)\|\|\_\{2\}^\{2\}\.\(7\)Here,ℱ​\(⋅,⋅\)\\mathcal\{F\}\(\\cdot,\\cdot\)denotes the output feature of the current sliding window,𝒜=\{α1,…,αl\}\\mathcal\{A\}=\\\{\\alpha\_\{1\},\.\.\.,\\alpha\_\{l\}\\\}and𝒯=\{T1,…,Tl\}\\mathcal\{T\}=\\\{\\textbf\{T\}\_\{1\},\.\.\.,\\textbf\{T\}\_\{l\}\\\}denote the set of scaling factors and the set of ternary weights to be solved via the softened ternarization defined in Equation[6](https://arxiv.org/html/2606.26650#S2.E6), and𝒜⋅𝒯=\{α1​T1,…,αl​Tl\}\\mathcal\{A\}\\cdot\\mathcal\{T\}=\\\{\\alpha\_\{1\}\\textbf\{T\}\_\{1\},\.\.\.,\\alpha\_\{l\}\\textbf\{T\}\_\{l\}\\\}\. During optimization, CAT\-Q forces the window\-wise outputs computed with ternary weights to match those computed with high\-precision weights under the same calibration inputs\. This relaxed formulation induces an implicit weight reconstruction, thereby further alleviating the difficulty of optimization\.

## 3Experiments

In this section, we conduct extensive experiments to validate the efficacy of our method, compare it with lots of related methods, and analyze the effect of key design choices\.

### 3\.1Setup

To ensure a comprehensive evaluation, we apply our method to a diverse set of LLMs, varying in architecture and model size\. Specifically, we select 7 dense models from the Llama2 and Qwen3 families\(Touvronet al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib17); Yanget al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib83)\), as well as 3 sparsely\-gated mixture of experts \(MoE\) models including Qwen3\-30B\-A3B, Qwen3\-235B\-A22B, and Ring\-flash\-2\.0 \(100B\-A6\.1B\)\(Teamet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib123)\), covering a wide model size range from 1\.7B to 235B parameters\. By default, we use 512 samples randomly selected from C4\(Raffelet al\.,[2020](https://arxiv.org/html/2606.26650#bib.bib7)\)for calibration, each with a length of 2048 tokens\. As for the choice of sliding window size, our method follows the default setting of SliderQuant\(Wanget al\.,[2026](https://arxiv.org/html/2606.26650#bib.bib119)\)\. Following common practices in QAT\-based ternarization research, all models are primarily evaluated in a zero\-shot setting on five widely adopted commonsense reasoning benchmarks: PIQA\(Bisket al\.,[2020](https://arxiv.org/html/2606.26650#bib.bib70)\), ARC\-Easy \(ARC\-e\) and ARC\-Challenge \(ARC\-c\)\(Clarket al\.,[2018](https://arxiv.org/html/2606.26650#bib.bib72)\), HellaSwag \(HS\)\(Zellerset al\.,[2019](https://arxiv.org/html/2606.26650#bib.bib71)\), and Winogrande \(WG\)\(Sakaguchiet al\.,[2021](https://arxiv.org/html/2606.26650#bib.bib73)\)\.

Table 1:Performance of different ternary LLMs produced by CAT\-Q in the PTQ regime and evaluated on five zero\-shot commonsense reasoning benchmarks\. Qwen3\-30B\-A3B, Ring\-flash\-2\.0 \(RF2\-100B\-A6\.1B\) and Qwen3\-235B\-A22B are MoE models, while the others are dense models\. The metric is accuracy \(%\)\.Table 2:Comparison of CAT\-Q and different QAT\-based ternarization methods under W1\.58A16 quantization\. \#Tokens denotes the number of training tokens for quantization\.
### 3\.2Counterpart Methods

We show the advantages of CAT\-Q, our proposed 1\.58\-bit PTQ method for LLMs, through a two\-part comparison\. First, we compare CAT\-Q with existing state\-of\-the\-art 1\.58\-bit quantization methods that rely on quantization\-aware training \(QAT\), including BitNet 1\.58\-bit v1\(Maet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib63)\), BitNet 1\.58\-bit v2\(Wanget al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib67)\), TernaryLLM\(Chenet al\.,[2024b](https://arxiv.org/html/2606.26650#bib.bib109)\), TriLM\(Kaushalet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib112)\)and Tequila\(Huanget al\.,[2026](https://arxiv.org/html/2606.26650#bib.bib113)\)\. Second, we compare CAT\-Q with some recent PTQ methods under their ultra\-low\-bit configurations: 2\-bit methods including GPTQ\(Frantaret al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib46)\), AWQ\(Linet al\.,[2024b](https://arxiv.org/html/2606.26650#bib.bib42)\), OmniQuant\(Shaoet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib4)\), EfficientQAT\(Chenet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib115)\)and SliderQuant\(Wanget al\.,[2026](https://arxiv.org/html/2606.26650#bib.bib119)\), as well as dual 1\-bit methods including BiLLM\(Huanget al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib36)\), PB\-LLM\(Shanget al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib37)\)and DB\-LLM\(Chenet al\.,[2024a](https://arxiv.org/html/2606.26650#bib.bib107)\)\.

### 3\.3Main Results

Ternary Quantization Results of CAT\-Q\.Table[2](https://arxiv.org/html/2606.26650#S3.T2)summarizes the results of CAT\-Q across 10 pre\-trained LLMs, covering diverse model architectures \(dense and MoE\), sizes \(1\.7B to 235B parameters\) and families \(Qwen3, Llama2, and Ring\-flash\-2\.0\)\. The performance degradation caused by quantization decreases as the model size increases, with Llama2\-70B exhibiting only a 3\.81% drop\. Meanwhile, MoE models show greater quantization sensitivity compared to dense models of similar scale, likely due to their fewer activated parameters\. Notably, we demonstrate for the first time that CAT\-Q can quantize LLMs having up to 235B parameters into leading ternary weight models within 60 hours on 8 A100\-80GB GPUs\. In addition, promising results are achieved with both W1\.58A8 and W1\.58A16 configurations\. These results highlight the broad applicability of CAT\-Q across diverse models and quantization settings\.

Table 3:Comparison of CAT\-Q and different QAT\-based ternarization methods under W1\.58A8 quantization\. \#Tokens denotes the number of training tokens for quantization\.Table 4:Comparison of CAT\-Q and recent PTQ methods with bit\-widths close to 1\.58\-bit\. PB\-LLM†, DB\-LLM†and BiLLM†use dual 1\-bit representations \(or say, binary residual\) and mixed\-precision quantization, so W1\* in them is close to or larger than W2A16 in terms of bit width\. EfficientQAT‡is actually a PTQ method as it uses pre\-trained LLMs and “QAT” is for re\-training\.Comparison with QAT\-based Ternarization Methods\.Table[2](https://arxiv.org/html/2606.26650#S3.T2)and Table[3](https://arxiv.org/html/2606.26650#S3.T3)compare the model performance of CAT\-Q with leading 1\.58\-bit QAT methods under W1\.58A16 and W1\.58A8 configurations\. Given that these QAT methods are limited to small model scales, we ensure a fair comparison by evaluating models of similar size\. We can observe that CAT\-Q offers a highly accurate and cost\-efficient solution for ternary quantization\. For instance, on LLMs with 1\.7B to 8B parameters, CAT\-Q with merely 512 calibration samples achieves competitive performance to BitNet 1\.58\-bit v1 and v2 families trained on 100B tokens, yielding a 100,000×\\timesreduction in training tokens\. Even compared to TernaryLLM\-8B, which uses 1T tokens for training guided by knowledge distillation \(KD\), our method still attains better overall accuracy\.

Table 5:Ablation of the complementarity ofδμ\\delta\_\{\\mu\},δα\\delta\_\{\\alpha\}andδΔ\\delta\_\{\\Delta\}in the learnable modulation \(LM\) component as well as the softened ternarization \(ST\) component under W1\.58A16 ternarization\.The baseline in the first row corresponds to SliderQuant without the proposed LM and ST components\.Comparison with Recent PTQ Methods\.In Table[4](https://arxiv.org/html/2606.26650#S3.T4), CAT\-Q is further compared with state\-of\-the\-art PTQ methods under their ultra\-low\-bit settings, which consist of 2\-bit methods and dual 1\-bit methods \(whose actual bit\-widths are around 2\-bit\)\. Despite operating at a lower 1\.58\-bit weight representation, in terms of the average accuracy, CAT\-Q is superior to these methods across both 7B and 70B model scales\. These results demonstrate that CAT\-Q achieves a substantially better efficiency\-accuracy trade\-off, providing larger compression and lower computational cost while maintaining better performance mostly\.

Table 6:Ablation of the softened ternarization process with varying choices of the calibration epoch ratioγ\\gammaallocated for differentiable ternarization\. The underlined results are for our default setting \(γ=0\.8\\gamma=0\.8\), and the best results are in bold\.Table 7:Ablation of the softened ternarization process with varying choices of the constants0s\_\{0\}\. The underlined results are for our default setting \(s0=30s\_\{0\}=30\), and the best results are in bold\.
### 3\.4Ablation Studies

To have a better understanding of the components and designs of CAT\-Q, we perform a series of ablative experiments\.

Role of Key Designs\.We first perform an ablation to evaluate the complementarity of our two core components: LM and ST\. Table[5](https://arxiv.org/html/2606.26650#S3.T5)summarizes the results\. Recall that LM leverages a composition of three learnable factorsδμ\\delta\_\{\\mu\},δα\\delta\_\{\\alpha\}, andδΔ\\delta\_\{\\Delta\}to mitigate the distributional weight misalignment and the ternary threshold\. Progressively incorporating these factors yields increasing performance gains, and the addition of ST significantly boosts the performance gain, validating the importance of the transition from differentiable ternarization to hard ternarization\. Coupling LM and ST leads to the best accuracy on both Qwen3\-4B and Llama2\-7B, confirming the effectiveness and synergy of our core designs\.

Choices of Calibration Epoch Ratioγ\\gamma\.The ratioγ\\gammacontrols the allocation of calibration epochs between the two stages of ST, namely a differentiable ternarization stage and a following hard ternarization stage\. As shown in Table[6](https://arxiv.org/html/2606.26650#S3.T6), CAT\-Q exhibits robust ternarization performance across all settings withγ≥0\.5\\gamma\\geq 0\.5\. A sweet spot is achieved atγ=0\.8\\gamma=0\.8, which we set as the default\. Empirically, we recommend settingγ\\gammabetween 0\.8 and 0\.9\. For ST, this range provides sufficient epochs for a smooth transition from the identity mapping to differentiable ternarization in the first stage while reserving adequate epochs for converging to a good hard ternarization solution in the second stage\. Althoughγ=1\.0\\gamma=1\.0remains viable, it leads to a relative accuracy drop, confirming the necessity of the hard ternarization stage\.

Table 8:Ablation of different strategies to determine the scaling factorα\\alphaand the thresholdΔ\\Delta\. We compare three strategies analyzed in the Method section, including static approximation \(SA\), direct learning \(DL\) and our learnable modulation \(LM\)\. The underlined results are for our default setting, and the best results are in bold\.Table 9:Ablation of LM with vs\. withoutμ\\mufor weight reconstruction namelyW≈α​T\+μ\\textbf\{W\}\\approx\\alpha\\textbf\{T\}\+\\muvs\.W≈α​T\\textbf\{W\}\\approx\\alpha\\textbf\{T\}\. The underlined results are for our default setting, and the best results are in bold\.Table 10:Ablation of CAT\-Q with learnable vs\. different hand\-crafted choices of the thresholdΔ\\Delta\. The underlined results are for our default setting, and the best results are in bold\.Choices of Constants0s\_\{0\}\.The constants0s\_\{0\}controls the maximum sharpness reached at the end of the differentiable ternarization stage in ST\. As shown in Table[7](https://arxiv.org/html/2606.26650#S3.T7), CAT\-Q achieves the best average performance withs0=30s\_\{0\}=30on both Qwen3\-4B and Llama2\-7B, which we set as the default\. Moderate values ofs0s\_\{0\}, such as 30 and 100, provide a good balance between smooth optimization and sufficient approximation to hard ternarization\. In contrast, a smalls0s\_\{0\}keeps the softened ternarization overly smooth and away from the target hard behavior, while an excessively larges0s\_\{0\}makes the transition reach the hard target too early and weakens the benefit of differentiable optimization\. This suggests that ST requires a gradual yet sufficiently sharp transition, and we recommend settings0s\_\{0\}around 30 in practice\.

Strategies to Determineα\\alphaandΔ\\Delta\.Note the LM component mitigates the distributional misalignment by adaptively transforming the weight distribution with learnable factorsδμ\\delta\_\{\\mu\},δα\\delta\_\{\\alpha\}, andδΔ\\delta\_\{\\Delta\}, making weights less sensitive to ternarization\. As evidenced by the lower weight reconstruction errors shown in Figure[2](https://arxiv.org/html/2606.26650#S2.F2), compared to SA and DL, LM better aligns the reconstructed weight distribution with the original weights\. The comparative results in Table[10](https://arxiv.org/html/2606.26650#S3.T10)further validate this conclusion, where LM consistently achieves the best results, underscoring its importance within our CAT\-Q\.

Weight Reconstruction Strategies\.In LM,μ\\muis used to redistribute the weights fromWtoW^\\hat\{\\textbf\{W\}\}but omitted in the reconstructionW≈α​T\\textbf\{W\}\\approx\\alpha\\textbf\{T\}for optimization\. As shown in Table[10](https://arxiv.org/html/2606.26650#S3.T10), this simplified form achieves slightly improved accuracy over retainingμ\\mu\(W≈α​T\+μ\\textbf\{W\}\\approx\\alpha\\textbf\{T\}\+\\mu\), while maintaining the hardware\-friendly deployment consistent with TWN\.

Learnable Threshold vs\. Hand\-crafted Thresholds\.In LM, the thresholdΔ\\Deltais optimized asΔ=δΔ​Δ0\\Delta=\\delta\_\{\\Delta\}\\Delta\_\{0\}whereδΔ\\delta\_\{\\Delta\}is a learnable factor\. We compare this learnable design against several hand\-crafted thresholds in Table[10](https://arxiv.org/html/2606.26650#S3.T10)\. The results confirm that the learnableΔ\\Deltaachieves the best performance, validating its effectiveness within our LM component\.

Table 11:Ablation of CAT\-Q with varying numbers of calibration samples\. The underlined results are for our default setting, and the best results are in bold\.Table 12:Ablation of CAT\-Q with varying quantization group sizes\. The underlined results are for our default setting, and the best results are in bold\.Table 13:Comparison of model deployment cases across different low\-bit formats withllama\.cppon an Intel Core i9\-13900KF CPU and an NVIDIA GeForce RTX 4090 GPU\. We report profiling results on both CPU and GPU for standard 4\-bit and 2\-bit models, together with the 1\.58\-bit model obtained by CAT\-Q, which is implemented asTQ1\_0on CPU andTQ2\_0on GPU\. We set the batch size to 1 and the generation length to 512 tokens\.Effect of the Number of Calibration Samples\.We also ablate the effect of the calibration sample count in Table[11](https://arxiv.org/html/2606.26650#S3.T11)\. The accuracy improves consistently as the sample count increases: on both Qwen3‑4B and Llama2‑7B, CAT\-Q attains the best performance when using 1024 samples\. We employ 512 samples as our default setting, which makes a favorable balance between ternarization cost and model accuracy\.

Effect of Group Size\.We finally ablate the impact of the quantization group size in Table[12](https://arxiv.org/html/2606.26650#S3.T12)\. The results indicate that smaller group sizes generally lead to better performance\. We use the group size of 128 as default, following recent 1\.58\-bit QAT\-based works TriLM and Tequila for fair comparisons with them\.

Acceleration in Model Deployment\.We further deploy ternary LLMs produced by CAT\-Q inllama\.cppusing its ternary formats, i\.e\.,TQ1\_0on CPU andTQ2\_0on GPU\. As shown in Table[13](https://arxiv.org/html/2606.26650#S3.T13), our ternary models consistently achieve lower weight memory and higher decoding throughput than standard 4\-bit and 2\-bit formats on both Qwen3\-4B and Llama2\-7B\. The efficiency gains hold on both the Intel Core i9\-13900KF CPU and the NVIDIA GeForce RTX 4090 GPU, showing that CAT\-Q can translate its 1\.58\-bit representation into practical inference efficiency improvements\.

In the Appendix, we further provide: \(1\) a summary of benchmark datasets and implementation details; \(2\) illustrations of the smooth transition functionf​\(⋅\)f\(\\cdot\)with varying sharpness choices ofss; \(3\) a pilot study of CAT\-Q on challenging mathematics and coding tasks; \(4\) a comparison of loss curves of CAT\-Q with vs\. without ST; \(5\) a more comprehensive comparison of different strategies for determiningα\\alphaandΔ\\Delta; \(6\) a discussion of limitations\.

## 4Related Work

Beyond the methods discussed earlier, in this section, we briefly review other relevant PTQ methods\.

There exist numerous PTQ methods\(Zhaoet al\.,[2019](https://arxiv.org/html/2606.26650#bib.bib34); Banneret al\.,[2019](https://arxiv.org/html/2606.26650#bib.bib22); Nagelet al\.,[2020](https://arxiv.org/html/2606.26650#bib.bib24); Liet al\.,[2021](https://arxiv.org/html/2606.26650#bib.bib29); Weiet al\.,[2022a](https://arxiv.org/html/2606.26650#bib.bib35)\)that focus on convolutional neural networks for computer vision tasks\. In contrast, LLMs have substantially larger model sizes and are harder to quantize, mainly due to the presence of outlier elements \(a small fraction of salient weights/activations with magnitudes significantly larger than the rest\) which induce severe quantization errors\. To tackle this problem, a variety of PTQ methods are proposed, among which mixed\-precision quantization is a widely adopted scheme\. It’s basic idea is to isolate outliers in high\-precision format \(e\.g\., FP16\) and quantize the remaining weights/activations into low\-precision representations \(e\.g\., INT8/INT4\)\. Representative examples include Q\-BERT\(Shenet al\.,[2020](https://arxiv.org/html/2606.26650#bib.bib41)\), LLM\.int8\(\)\(Dettmerset al\.,[2022](https://arxiv.org/html/2606.26650#bib.bib39)\), SpQR\(Dettmerset al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib51)\), QUIK\(Ashkbooset al\.,[2024a](https://arxiv.org/html/2606.26650#bib.bib57)\)and SqueezeLLM\(Kimet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib52)\)\. However, they are not hardware\-efficient for deployment\. Instead, Outlier Suppression\(Weiet al\.,[2022b](https://arxiv.org/html/2606.26650#bib.bib40)\)combines LayerNorm migration and token\-wise clipping to make activations more amenable to 8\-bit quantization\. ZeroQuant\(Yaoet al\.,[2022](https://arxiv.org/html/2606.26650#bib.bib62)\)adopts a fine\-grained INT8 quantization scheme consisting of group\-wise quantization for weights and token\-wise quantization for activations\. GPTQ\(Frantaret al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib46)\)leverages approximate second\-order Hessian matrices to suppress the impact of weight outliers\. Another line of PTQ research for LLMs relies on equivalent transformations to mitigate the quantization difficulty of outliers\. For instance, SmoothQuant\(Xiaoet al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib59)\)applies channel\-wise smoothing, QLLM\(Liuet al\.,[2024b](https://arxiv.org/html/2606.26650#bib.bib56)\)adopts channel\-split\-merge, QuIP\(Cheeet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib50)\)employs incoherence processing and QuaRot\(Ashkbooset al\.,[2024b](https://arxiv.org/html/2606.26650#bib.bib5)\)uses random rotation matrices\. Subsequent variants improve them by learning transformations\(Shaoet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib4); Linet al\.,[2024b](https://arxiv.org/html/2606.26650#bib.bib42),[a](https://arxiv.org/html/2606.26650#bib.bib55); Tsenget al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib49); Liuet al\.,[2025a](https://arxiv.org/html/2606.26650#bib.bib60); Sunet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib61)\)\. These PTQ methods primarily consider basic low\-bit settings and perform poorly in ultra\-low\-bit settings \(e\.g\., 3\-bit quantization and 2\-bit quantization\)\. Our method differs with them in focus, formulation and deployment\.

There also exist some works\(Huanget al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib36); Shanget al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib37); Liet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib122)\)tailored for binary quantization of LLMs\. As another standalone research direction, binary quantization is more challenging than the ternary quantization \(this paper’s focus\)\. In formulation, these methods usually adopt mixed\-precision quantization and binary residuals \(dual 1\-bit representations\) to avoid performance collapse, following earlier binarization techniques developed for vision models\(Linet al\.,[2017](https://arxiv.org/html/2606.26650#bib.bib120); Guoet al\.,[2017](https://arxiv.org/html/2606.26650#bib.bib121)\)\.

## 5Conclusion

In this paper, we introduce CAT\-Q, a cost\-efficient and accurate quantization method for ternarizing the weights of LLMs in the post\-training regime\. It consists of two core components, learnable modulation and softened ternarization, which are coupled into a sliding\-layer quantization pipeline to learn scaling factors and ternary thresholds across multiple layers jointly\. Extensive experiments show the advantages of CAT\-Q over existing methods that rely on QAT\. Our method makes it substantially easier to obtain and deploy high\-performing ternary weight LLMs in practice\.

## Impact Statement

This paper presents work whose goal is to advance the field of machine learning\. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here\.

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- R\. Zellers, A\. Holtzman, Y\. Bisk, A\. Farhadi, and Y\. Choi \(2019\)HellaSwag: can a machine really finish your sentence?\.InACL,Cited by:[Appendix A](https://arxiv.org/html/2606.26650#A1.p4.1),[§3\.1](https://arxiv.org/html/2606.26650#S3.SS1.p1.1)\.
- A\. Zhang, Y\. Tay, S\. Zhang, A\. Chan, A\. T\. Luu, S\. C\. Hui, and J\. Fu \(2021\)Beyond fully\-connected layers with quaternions: parameterization of hypercomplex multiplications with 1/n parameters\.InICLR,Cited by:[§1](https://arxiv.org/html/2606.26650#S1.p1.1)\.
- W\. Zhang, L\. Hou, Y\. Yin, L\. Shang, X\. Chen, X\. Jiang, and Q\. Liu \(2020\)TernaryBERT: distillation\-aware ultra\-low bit bert\.InEMNLP,Cited by:[§2\.1](https://arxiv.org/html/2606.26650#S2.SS1.p1.8)\.
- R\. Zhao, Y\. Hu, J\. Dotzel, C\. De Sa, and Z\. Zhang \(2019\)Improving neural network quantization without retraining using outlier channel splitting\.InICML,Cited by:[§4](https://arxiv.org/html/2606.26650#S4.p2.1)\.
- C\. Zhu, S\. Han, H\. Mao, and W\. J\. Dally \(2017\)Trained ternary quantization\.InICLR,Cited by:[§1](https://arxiv.org/html/2606.26650#S1.p4.1),[§2\.2](https://arxiv.org/html/2606.26650#S2.SS2.p1.17)\.

## Appendix

- •Section[A](https://arxiv.org/html/2606.26650#A1): Datasets used in experiments\.
- •Section[B](https://arxiv.org/html/2606.26650#A2): Implementation details of CAT\-Q\.
- •Section[C](https://arxiv.org/html/2606.26650#A3): Illustrations of the smooth transition functionf​\(⋅\)f\(\\cdot\)with varying sharpness\.
- •Section[D](https://arxiv.org/html/2606.26650#A4): Comparison of loss curves of CAT\-Q with vs\. without the ST component\.
- •Section[E](https://arxiv.org/html/2606.26650#A5): A pilot study of CAT\-Q on challenging mathematics and coding tasks\.
- •Section[F](https://arxiv.org/html/2606.26650#A6): A more comprehensive comparison of different strategies for determiningα\\alphaandΔ\\Delta\.
- •Section[G](https://arxiv.org/html/2606.26650#A7): Discussion of limitations\.

## Appendix ADatasets Used in Experiments

C4\(Raffelet al\.,[2020](https://arxiv.org/html/2606.26650#bib.bib7)\)\(Colossal Clean Crawled Corpus\) is a large\-scale corpus widely adopted for language modeling research\. It consists of approximately 156 billion tokens collected from web documents and processed through extensive filtering and cleaning procedures based on Common Crawl\.

PIQA\(Bisket al\.,[2020](https://arxiv.org/html/2606.26650#bib.bib70)\)is a benchmark for physical commonsense reasoning, comprising about 16,000 training instances and 3,000 validation instances\. Each example is presented as a multiple\-choice question with two candidate solutions, where exactly one option is correct\.

ARC\(Clarket al\.,[2018](https://arxiv.org/html/2606.26650#bib.bib72)\)contains 7,787 multiple\-choice science questions at the grade\-school level, designed to promote progress in question answering and reasoning\. The dataset is divided into an Easy set and a Challenge set, with the latter including questions that are not solvable by simple retrieval or co\-occurrence–based methods\.

HellaSwag\(Zellerset al\.,[2019](https://arxiv.org/html/2606.26650#bib.bib71)\)includes roughly 70,000 training samples and 10,000 validation samples\. It evaluates commonsense reasoning by requiring models to select the most plausible continuation of a given context, which is constructed from crowdsourced activity descriptions and image captions\.

Winogrande\(Sakaguchiet al\.,[2021](https://arxiv.org/html/2606.26650#bib.bib73)\)consists of approximately 44,000 instances formulated as binary\-choice fill\-in\-the\-blank problems\. The task requires selecting the correct option to complete a sentence, emphasizing coreference resolution and commonsense reasoning capabilities\.

MATH\-500\(Hendryckset al\.,[2021](https://arxiv.org/html/2606.26650#bib.bib76)\)comprises 500 challenging competition\-level mathematics problems sampled from the MATH dataset\. These problems span various topics such as algebra, geometry, number theory, and probability, and are designed to test a model’s ability to perform complex mathematical reasoning and generate step\-by\-step solutions with rigorous intermediate derivations\.

Omni\-MATH\(Gaoet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib124)\)is a large\-scale benchmark for olympiad\-level mathematical reasoning, containing 4,428 problems collected from diverse national and international mathematics competitions\. The dataset covers a broad range of advanced topics and is designed to evaluate models on difficult multi\-step problem solving beyond standard school\-level math benchmarks, posing higher demands on abstraction, symbolic manipulation, and long\-chain reasoning\.

GSM8K\(Cobbeet al\.,[2021](https://arxiv.org/html/2606.26650#bib.bib79)\)is a dataset of 8,792 high\-quality, linguistically diverse grade school math word problems created by human problem writers\. The dataset is segmented into 7,473 training problems and 1,319 test problems, each requiring multi\-step reasoning and basic arithmetic operations to solve, while emphasizing the ability to translate natural language descriptions into executable reasoning procedures\.

HumanEval\+\(Liuet al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib77)\)is an extension of the HumanEval dataset, consisting of 164 original programming problems designed to assess the functional correctness of code generated by language models\. Each problem includes a function signature, a docstring specifying the intended functionality, and multiple test cases for evaluation, enabling a more reliable assessment of whether generated programs satisfy the expected behavior\.

MBPP\+\(Liuet al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib77)\)is an augmented version of the Mostly Basic Programming Problems \(MBPP\) dataset, comprising approximately 378 crowd\-sourced Python programming tasks\. Each task includes a natural language description, a reference solution, and three test cases, aiming to evaluate models’ abilities in basic programming and problem\-solving across diverse everyday coding scenarios\.

## Appendix BImplementation Details of CAT\-Q

### B\.1Hyper\-parameter Settings

Table A:The detailed hyper\-parameter settings of CAT\-Q\.Table[A](https://arxiv.org/html/2606.26650#A2.T1)summarizes the complete hyper\-parameter configuration used in our experiments\. We perform group\-wise ternary quantization with a fixed group size of 128 and use a calibration set constructed from C4, consisting of 512 samples with 2048 tokens per sample\. The table reports the initialization, including the ternary thresholdΔ0=0\.5\\Delta\_\{0\}=0\.5, the constants0=30s\_\{0\}=30and the differentiable ternarization ratioγ=0\.8\\gamma=0\.8\. It also summarizes the optimization\-related settings of the batch size, the optimizer, the number of training epochs, and the learning rate schedule\. Unless otherwise stated, all experiments follow this configuration\.

### B\.2Quantization Details

We follow the ternary quantization formulation introduced in the main text and do not restate the derivation here\. This section only summarizes implementation details that are not explicitly covered in the main paper\. In practice, we adopt a group\-wise quantization scheme, where the weight tensor is flattened and partitioned into non\-overlapping groups with a fixed group size ofg=128g=128, and all related statistics and parameters are handled independently at the group level\. After group\-wise normalization and ternarization as described in the main paper, the original weights are approximated as𝐖≈α​𝐓\\mathbf\{W\}\\approx\\alpha\\mathbf\{T\}withoutμ\\mu\. At deployment time, the ternary weights remain in\{−1,0,\+1\}\\\{\-1,0,\+1\\\}, and only the group\-wise scaling factorα\\alphais required, which can be absorbed into surrounding linear operations\. This zero\-point–free design preserves the hardware\-friendly properties of standard ternary quantization and is applicable to both weight\-only and weight–activation quantization settings\. For experiments involving activation quantization, all intermediate activations are quantized following the same calibration procedure, except for the Softmax output probability vectors, which are kept in full precision for numerical stability\. When handling MoE models, the routing modules used for expert selection in the MLP are excluded from quantization, as they constitute a negligible fraction of the total parameters while being critical to expert assignment\.

![Refer to caption](https://arxiv.org/html/2606.26650v1/x3.png)Figure A:Illustrative output curves of our proposed smooth transition functionf​\(⋅\)f\(\\cdot\)with varying sharpness choices ofss\. Asssincreases,f​\(⋅\)f\(\\cdot\)gradually transitions from a finite sequence of continuous mappings with increasing sharpness to a near\-discrete ternary weight output in terms of the high\-precision weight input, approximating a hard ternarization solution\.

## Appendix CIllustrations of the Smooth Transition Functionf​\(⋅\)f\(\\cdot\)with Varying Sharpness

Figure[A](https://arxiv.org/html/2606.26650#A2.F1)visualizes the smooth transition function defined in Section[2\.3](https://arxiv.org/html/2606.26650#S2.SS3)\(Equation[5](https://arxiv.org/html/2606.26650#S2.E5)\) under different values of the sharpness parameterss, illustrating how the softened ternarization state evolves asssincreases\. Here,sscontrols the instantaneous sharpness of the transition function during training, whiles0s\_\{0\}denotes the final sharpness value reached at the end of the differentiable ternarization stage\. As discussed in Section[3\.4](https://arxiv.org/html/2606.26650#S3.SS4)with Table[7](https://arxiv.org/html/2606.26650#S3.T7), an appropriates0s\_\{0\}should make the softened ternarization sufficiently close to hard ternarization while preserving a smooth optimization trajectory\. The visualization provides an intuitive explanation for this trade\-off: small values ofsslead to an overly smooth transition, whereas excessively large values make the function nearly hard\. This supports our default choice ofs0=30s\_\{0\}=30in CAT\-Q\.

![Refer to caption](https://arxiv.org/html/2606.26650v1/x4.png)\(a\)1s​t1^\{st\}sliding window
![Refer to caption](https://arxiv.org/html/2606.26650v1/x5.png)\(b\)5t​h5^\{th\}sliding window
![Refer to caption](https://arxiv.org/html/2606.26650v1/x6.png)\(c\)9t​h9^\{th\}sliding window
![Refer to caption](https://arxiv.org/html/2606.26650v1/x7.png)\(d\)13t​h13^\{th\}sliding window
![Refer to caption](https://arxiv.org/html/2606.26650v1/x8.png)\(e\)17t​h17^\{th\}sliding window
![Refer to caption](https://arxiv.org/html/2606.26650v1/x9.png)\(f\)20t​h20^\{th\}sliding window

Figure B:Comparison of the calibration loss curves of CAT\-Q with vs\. without the softened ternarization \(ST\) component\. We use the1s​t,5t​h,9t​h,13t​h,17t​h1^\{st\},5^\{th\},9^\{th\},13^\{th\},17^\{th\}and20t​h20^\{th\}sliding windows of our method with Qwen3\-4B for illustrative comparisons\. By introducing the ST component, although the loss curves of CAT\-Q exhibit a different optimization trajectory trend compared to those of merely using the learnable modulation \(LM\) component, it consistently converges to substantially lower loss values\. This behavior supports the core insight of the ST component, which enables smoother optimization and mitigates the adverse effect of early\-epoch hard ternarization\.
## Appendix DComparison of Loss Curves of CAT\-Q with vs\. without the ST Component

Figure[B](https://arxiv.org/html/2606.26650#A3.F2)illustrates the evolution of quantization loss across training steps under the proposed sliding\-layer ternarization optimization paradigm \(Section[2\.4](https://arxiv.org/html/2606.26650#S2.SS4)\)\. When softened ternarization \(ST\) is employed, the quantization loss increases smoothly during training and converges to a substantially lower final value than the baseline without ST\. This behavior stems from the gradual annealing of ST from an identity\-like mapping to hard ternarization, which mitigates early quantization\-induced perturbations and preserves stable gradient propagation\. In contrast, applying hard ternarization throughout training introduces strong non\-smoothness from the outset, resulting in inferior convergence and higher residual quantization error\. Overall, ST leads to smoother optimization trajectories and consistently improved convergence, validating the effectiveness of the proposed softened ternarization strategy\.

## Appendix EA Pilot Study of CAT\-Q on Challenging Mathematics and Coding Tasks

We further evaluate CAT\-Q on challenging mathematics and coding tasks, including Math\-500\(Hendryckset al\.,[2021](https://arxiv.org/html/2606.26650#bib.bib76)\), GSM8K\(Cobbeet al\.,[2021](https://arxiv.org/html/2606.26650#bib.bib79)\), Omni\-MATH\(Gaoet al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib124)\), HumanEval\+\(Liuet al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib77)\), and MBPP\+\(Liuet al\.,[2023](https://arxiv.org/html/2606.26650#bib.bib77)\)\. As shown in Table[B](https://arxiv.org/html/2606.26650#A5.T2), on these challenging tasks, directly applying CAT\-Q under W1\.58A16 suffers from serious performance degradation\. We study the underlying causes of this issue and propose CAT\-Q\+ in a separate technical report to be released in the near future\. CAT\-Q\+ introduces a new calibration data generation strategy while retaining the original CAT\-Q quantization method, resulting in substantially improved performance\.

Table B:Pilot results on challenging mathematics and coding tasks under W1\.58A16 ternarization\. CAT\-Q\+ denotes an enhanced variant of CAT\-Q under the same quantization setting but with a new calibration data generation strategy\.
## Appendix FA More Comprehensive Comparison of Different Strategies for Determiningα\\alphaandΔ\\Delta\.

This section visualizes the weight reconstruction errors under different strategies for determining the scaling factorα\\alphaand the thresholdΔ\\Deltaacross multiple models and layers\. Specifically, we show the results for the4t​h,15t​h4^\{th\},15^\{th\}and30t​h30^\{th\}layers of Qwen3\-4B \(Figures[C](https://arxiv.org/html/2606.26650#A7.F3)to[E](https://arxiv.org/html/2606.26650#A7.F5)\) and Llama2\-7B \(Figures[F](https://arxiv.org/html/2606.26650#A7.F6)to[H](https://arxiv.org/html/2606.26650#A7.F8)\), as well as the MLP gate, up, and down projections of Qwen3\-30B\-A3B \(Figures[I](https://arxiv.org/html/2606.26650#A7.F9)to[K](https://arxiv.org/html/2606.26650#A7.F11)\), where each visualization includes multiple experts\. Despite differences in model scale, architecture, and layer type, all figures exhibit highly consistent trends\. Across all settings, static approximation \(used in BitNet 1\.58\-bit families\(Maet al\.,[2024](https://arxiv.org/html/2606.26650#bib.bib63); Wanget al\.,[2025](https://arxiv.org/html/2606.26650#bib.bib67)\)\) results in the largest reconstruction errors, indicating limited representational flexibility under fixed quantization statistics\. Direct learning \(initialized with the above static approximation used in BitNet 1\.58\-bit families\) generally reduces the overall reconstruction error compared to static approximation\. However, it also introduces noticeable outliers, with certain weight groups exhibiting errors even larger than those produced by static approximation\. In contrast, the proposed learnable modulation consistently achieves the lowest reconstruction errors with significantly reduced variance, demonstrating both improved accuracy and stability\. These observations consistently validate the effectiveness and robustness of our learnable modulation strategy across different models, layers, and expert configurations\.

## Appendix GDiscussion of Limitations

Though we have provided two real\-world 1\.58\-bit cases and shown their inference advantages to 2\-bit and 4\-bit formats, the current deployments remain under\-optimized\. BitNet’s public 1\.58\-bit GPU and CPU kernels are limited to its own models\. That is, optimized kernel support for diverse 1\.58\-bit LLMs remains lacking\. Besides, although our CAT\-Q is a universal, low\-cost and accurate 1\.58\-bit PTQ method, there remains considerable room to reduce the performance gap to match the FP16 baseline, especially on complex reasoning tasks that are largely underexplored in LLM quantization research\.

![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_4-self_attn.q_proj_nolegend.png)\(a\)self\_attn\.q\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_4-self_attn.k_proj_nolegend.png)\(b\)self\_attn\.k\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_4-self_attn.v_proj_nolegend.png)\(c\)self\_attn\.v\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_4-self_attn.o_proj_nolegend.png)\(d\)self\_attn\.o\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_4-mlp.up_proj_nolegend.png)\(e\)mlp\.up\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_4-mlp.down_proj_nolegend.png)\(f\)mlp\.down\_proj

Figure C:Comparison of weight reconstruction errors with the scaling factorα\\alphaand the thresholdΔ\\Deltadetermined by static approximation \(blue dots\), direct learning \(orange dots\) and our learnable modulation \(green dots\)\. Under the same ternarization setting, we use the4t​h4^\{th\}layer of Qwen3\-4B for an illustration\.![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_15-self_attn.q_proj_nolegend.png)\(a\)self\_attn\.q\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_15-self_attn.k_proj_nolegend.png)\(b\)self\_attn\.k\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_15-self_attn.v_proj_nolegend.png)\(c\)self\_attn\.v\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_15-self_attn.o_proj_nolegend.png)\(d\)self\_attn\.o\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_15-mlp.up_proj_nolegend.png)\(e\)mlp\.up\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_15-mlp.down_proj_nolegend.png)\(f\)mlp\.down\_proj

Figure D:Comparison of weight reconstruction errors with the scaling factorα\\alphaand the thresholdΔ\\Deltadetermined by static approximation \(blue dots\), direct learning \(orange dots\) and our learnable modulation \(green dots\)\. Under the same ternarization setting, we use the15t​h15^\{th\}layer of Qwen3\-4B for an illustration\.![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_30-self_attn.q_proj_nolegend.png)\(a\)self\_attn\.q\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_30-self_attn.k_proj_nolegend.png)\(b\)self\_attn\.k\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_30-self_attn.v_proj_nolegend.png)\(c\)self\_attn\.v\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_30-self_attn.o_proj_nolegend.png)\(d\)self\_attn\.o\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_30-mlp.up_proj_nolegend.png)\(e\)mlp\.up\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-4b-compressed/loss-layer_30-mlp.down_proj_nolegend.png)\(f\)mlp\.down\_proj

Figure E:Comparison of weight reconstruction errors with the scaling factorα\\alphaand the thresholdΔ\\Deltadetermined by static approximation \(blue dots\), direct learning \(orange dots\) and our learnable modulation \(green dots\)\. Under the same ternarization setting, we use the30t​h30^\{th\}layer of Qwen3\-4B for an illustration\.![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_4-self_attn.q_proj_nolegend.png)\(a\)self\_attn\.q\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_4-self_attn.k_proj_nolegend.png)\(b\)self\_attn\.k\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_4-self_attn.v_proj_nolegend.png)\(c\)self\_attn\.v\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_4-self_attn.o_proj_nolegend.png)\(d\)self\_attn\.o\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_4-mlp.up_proj_nolegend.png)\(e\)mlp\.up\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_4-mlp.down_proj_nolegend.png)\(f\)mlp\.down\_proj

Figure F:Comparison of weight reconstruction errors with the scaling factorα\\alphaand the thresholdΔ\\Deltadetermined by static approximation \(blue dots\), direct learning \(orange dots\) and our learnable modulation \(green dots\)\. Under the same ternarization setting, we use the4t​h4^\{th\}layer of Llama2\-7B for an illustration\.![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_15-self_attn.q_proj_nolegend.png)\(a\)self\_attn\.q\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_15-self_attn.k_proj_nolegend.png)\(b\)self\_attn\.k\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_15-self_attn.v_proj_nolegend.png)\(c\)self\_attn\.v\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_15-self_attn.o_proj_nolegend.png)\(d\)self\_attn\.o\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_15-mlp.up_proj_nolegend.png)\(e\)mlp\.up\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_15-mlp.down_proj_nolegend.png)\(f\)mlp\.down\_proj

Figure G:Comparison of weight reconstruction errors with the scaling factorα\\alphaand the thresholdΔ\\Deltadetermined by static approximation \(blue dots\), direct learning \(orange dots\) and our learnable modulation \(green dots\)\. Under the same ternarization setting, we use the15t​h15^\{th\}layer of Llama2\-7B for an illustration\.![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_30-self_attn.q_proj_nolegend.png)\(a\)self\_attn\.q\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_30-self_attn.k_proj_nolegend.png)\(b\)self\_attn\.k\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_30-self_attn.v_proj_nolegend.png)\(c\)self\_attn\.v\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_30-self_attn.o_proj_nolegend.png)\(d\)self\_attn\.o\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_30-mlp.up_proj_nolegend.png)\(e\)mlp\.up\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/llama2-7b-compressed/loss-layer_30-mlp.down_proj_nolegend.png)\(f\)mlp\.down\_proj

Figure H:Comparison of weight reconstruction errors with the scaling factorα\\alphaand the thresholdΔ\\Deltadetermined by static approximation \(blue dots\), direct learning \(orange dots\) and our learnable modulation \(green dots\)\. Under the same ternarization setting, we use the30t​h30^\{th\}layer of Llama2\-7B for an illustration\.![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.20.gate_proj_nolegend.png)\(a\)mlp\.experts\.20\.gate\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.40.gate_proj_nolegend.png)\(b\)mlp\.experts\.40\.gate\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.60.gate_proj_nolegend.png)\(c\)mlp\.experts\.60\.gate\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.80.gate_proj_nolegend.png)\(d\)mlp\.experts\.80\.gate\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.100.gate_proj_nolegend.png)\(e\)mlp\.experts\.100\.gate\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.120.gate_proj_nolegend.png)\(f\)mlp\.experts\.120\.gate\_proj

Figure I:Comparison of weight reconstruction errors with the scaling factorα\\alphaand the thresholdΔ\\Deltadetermined by static approximation \(blue dots\), direct learning \(orange dots\) and our learnable modulation \(green dots\)\. Under the same ternarization setting, we use the15t​h15^\{th\}layer \(gate\_proj\) of Qwen3\-30B\-A3B, where six experts are uniformly sampled from the total 128 experts for an illustration\.![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.20.up_proj_nolegend.png)\(a\)mlp\.experts\.20\.up\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.40.up_proj_nolegend.png)\(b\)mlp\.experts\.40\.up\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.60.up_proj_nolegend.png)\(c\)mlp\.experts\.60\.up\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.80.up_proj_nolegend.png)\(d\)mlp\.experts\.80\.up\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.100.up_proj_nolegend.png)\(e\)mlp\.experts\.100\.up\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.120.up_proj_nolegend.png)\(f\)mlp\.experts\.120\.up\_proj

Figure J:Comparison of weight reconstruction errors with the scaling factorα\\alphaand the thresholdΔ\\Deltadetermined by static approximation \(blue dots\), direct learning \(orange dots\) and our learnable modulation \(green dots\)\. Under the same ternarization setting, we use the15t​h15^\{th\}layer \(up\_proj\) of Qwen3\-30B\-A3B, where six experts are uniformly sampled from the total 128 experts for an illustration\.![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.20.down_proj_nolegend.png)\(a\)mlp\.experts\.20\.down\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.40.down_proj_nolegend.png)\(b\)mlp\.experts\.40\.down\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.60.down_proj_nolegend.png)\(c\)mlp\.experts\.60\.down\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.80.down_proj_nolegend.png)\(d\)mlp\.experts\.80\.down\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.100.down_proj_nolegend.png)\(e\)mlp\.experts\.100\.down\_proj
![Refer to caption](https://arxiv.org/html/2606.26650v1/Figures/learned_factors/qwen3-30b-moe-compressed/loss-layer_15-mlp.experts.120.down_proj_nolegend.png)\(f\)mlp\.experts\.120\.down\_proj

Figure K:Comparison of weight reconstruction errors with the scaling factorα\\alphaand the thresholdΔ\\Deltadetermined by static approximation \(blue dots\), direct learning \(orange dots\) and our learnable modulation \(green dots\)\. Under the same ternarization setting, we use the15t​h15^\{th\}layer \(down\_proj\) of Qwen3\-30B\-A3B, where six experts are uniformly sampled from the total 128 experts for an illustration\.