Title: PolarQuant: Leveraging Polar Transformation for Key Cache Quantization and Decoding Acceleration
Venue: NeurIPS 2025
Accepted: yes
Keywords: Quantization

KV Cache

LLM Inference
Abstract: The increasing demand for long-context generation has made the KV cache in large language models a bottleneck in memory consumption. Quantizing the cache to lower bit widths is an effective way to reduce memory costs; however, previous methods struggle with key cache quantization due to outliers, resulting in suboptimal performance. We propose a novel quantization approach PolarQuant, which provides a new perspective for key cache quantization and efficiently addresses the outlier dilemma. We observe that the distribution of the key states reveals well-structured patterns under polar transformation. Outliers generally appear in only one of the two dimensions, which are rotated together by a specific angle when rotary position embeddings are applied. When represented as two-dimensional vectors, these dimensions exhibit well-organized patterns, with radii and angles smoothly distributed in polar space. This alleviates the channel-wise outliers, making them well-suited for key cache quantization. PolarQuant divides key vectors into groups of two-dimensional sub-vectors, encoding them as the quantized radius and the polar angle, rather than quantizing original key vectors directly. PolarQuant achieves the superior efficiency in KV cache quantization and accelerates the decoding process by turning the query-key inner product into a table lookup, all while maintaining the downstream performance of full-precision models.
Our code is available at https://github.com/ericshwu/PolarQuant.

PolarQuant: Leveraging Polar Transformation for Key Cache Quantization and Decoding Acceleration

Abstract

The increasing demand for long-context generation has made the KV cache in large language models a bottleneck in memory consumption. Quantizing the cache to lower bit widths is an effective way to reduce memory costs; however, previous methods struggle with key cache quantization due to outliers, resulting in suboptimal performance. We propose a novel quantization approach PolarQuant, which provides a new perspective for key cache quantization and efficiently addresses the outlier dilemma. We observe that the distribution of the key states reveals well-structured patterns under polar transformation. Outliers generally appear in only one of the two dimensions, which are rotated together by a specific angle when rotary position embeddings are applied. When represented as two-dimensional vectors, these dimensions exhibit well-organized patterns, with radii and angles smoothly distributed in polar space. This alleviates the channel-wise outliers, making them well-suited for key cache quantization. PolarQuant divides key vectors into groups of two-dimensional sub-vectors, encoding them as the quantized radius and the polar angle, rather than quantizing original key vectors directly. PolarQuant achieves the superior efficiency in KV cache quantization and accelerates the decoding process by turning the query-key inner product into a table lookup, all while maintaining the downstream performance of full-precision models. Our code is available at https://github.com/ericshwu/PolarQuant .

1 Introduction

Large language models (LLMs) have achieved remarkable success across a wide range of realworld applications. As these models evolve, the demand for enhanced long-context capabilities also increases, especially in tasks like contextual retrieval in question answering [17] and reasoning chains generation for complex reflection and decision-making [21, 20]. However, a significant challenge in developing long-context LLMs is the rising memory cost associated with increasing context lengths, which hinders both their practical deployment and further research.

The attention mechanism [4] in LLMs 3 is a major source of computational overhead and memory consumption, which increase rapidly with context length. To reduce this cost, Key-Value cache (KV cache) is a common strategy, which stores and reuses keys and values for generation to avoid the

x

y

O

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L

O

𝜽 = 𝟎

𝜽 = 𝝅

𝜽 =

𝟑𝝅

𝟐

𝜽 =

𝝅

𝟒

𝜽 =

𝟑𝝅

𝟒

𝜽 =

𝟓𝝅

𝟒

𝜽 =

𝟕𝝅

𝟒

𝜽 =

𝝅

𝟐

0

1

2

3

001

011

101

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01

01

11

𝑲𝒏 𝟐𝒋 + 𝟏

𝑲𝒏 𝒋 ,

(𝝆,𝜽)

(𝒓, 𝒕)

PolarQuant

Magnitude

𝒕 = 𝟑

𝒓 = 𝟐

(a)	Llama-3.1-8B-Instruct

Layer	0	Head	0	Key	Cache

Structured	Circular	Pattern

(c)	Quantization	Schema

Channel Magnitude as Y-AxisCoordinate

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256.

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redundant computation. Nevertheless, as the context length increase, the memory required for KV cache can surpass that of the model weights, making it the dominant factor in overall memory usage.

Many solutions have been proposed to reduce the memory cost of KV cache. Some studies introduce memory-efficient attention modules, such as MQA [25], GQA [2] and MLA [9]. While promising, these module architectures require training LLMs from scratch, which limits their applicability. Another research line focuses on the reduction of KV cache size in a compatible manner with pretrained LLMs. This includes techniques like KV cache eviction [7, 15, 36], which identifies and drops unimportant tokens from the cache , and KV cache quantization [12-14, 19, 37].

This paper focuses on the key cache quantization, which converts the floating-point key cache into lowbit integers to reduce memory usage. In general, key cache quantization is more challenging than value cache due to the presence of channel-wise distributed outliers. Prior studies [13, 19] have highlighted the widespread existence of such outliers in key cache. As shown in Figure 1(a), the key states exhibit larger activations along certain channel dimensions, making token-wise quantization difficult. To address this issue, KIVI [19] proposes a channel-wise quantization strategy that groups and quantizes key elements along the channel dimensions. Building upon this perspective, KVQuant [13] further identifies RoPE as the primary source responsible of the outliers observed in the key cache. The rotation operations in RoPE disturb the magnitude consistency, making accurate quantization complicated. To mitigate this, KVQuant proposes quantizing the keys before applying RoPE, which is described as pre-RoPE quantization. Promising as it is, this approach requires on-the-fly RoPE computation, which consequently introduces potential computational overhead. In this work, we aim to preserve the benefits of pre-RoPE quantization in reducing approximation errors while eliminating redundant computations at each generation step. We propose a polar transformation perspective on handling outliers in the key cache, and effectively address the dilemma in 2D polar coordinates.

KVQuant [13] observes a clear and structured pattern in pre-RoPE key activations: channel-wise magnitude are highly consistent. Recall that RoPE operates a rotation to every two-dimensional

sub-vector within the key using an orthogonal 2 × 2 rotary matrices. 4 Since rotation is a magnitudepreserving transformation, these 2D sub-vectors inherit the magnitude characteristics seen in the pre-RoPE case. As shown in Figure 1(b), they form well-structured circular patterns when analyzed in 2D polar coordinates. By encoding each sub-vector as its corresponding radius ρ and polar angle θ , the entire key vector can be represented as a collection of all radii and angles. This transformation effectively mitigates outliers, as both the radii and polar angles become smoothly distributed. Building on this, we propose a novel quantization method, PolarQuant, which significantly simplifies the quantization of the key cache. PolarQuant reduces the problem of quantizing key vectors to asymmetrically quantizing ρ and θ into an r -bit and an t -bit integer. Intuitively, PolarQuant defines 2 r + t distinct regions based on 2 r angles and 2 t radii. Each sub-vector is then encoded by the index of the region it belongs to. Figure 1(c) illustrates PolarQuant for r = 2 and t = 3 .

PolarQuant achieves superior quantization effectiveness and efficiency over previous methods. On one hand, polar transformation enables smoother distributions of radii and angles, which alleviates the burden of channel-wise quantization outliers. The superior performance on downstream tasks further demonstrates PolarQuant's superiority in quantization error reduction.

On the other hand, PolarQuant offers a brand new perspective on key cache quantization, which enables a novel decoding acceleration method. Unlike pre-RoPE quantization like [13], PolarQuant eliminates the overhead of RoPE recomputation. In the attention mechanism, it replaces the standard query-key multiplication with inner products between two-dimensional query subvectors and a quantized polar coordinate representation of key sub-vectors, which have finite and deterministic states. This transforms matrix multiplication to a table lookup, greatly speeding up attention computation. Our contributions are threefold:

(1) We introduce polar transformation for key cache quantization for the first time and derive PolarQuant, a novel and efficient quantization approach;

(2) We propose a new decoding acceleration algorithm as a natural byproduct of PolarQuant. We implement custom Triton kernels to perform fused dequantization and query-key multiplication, which achieves up to 3.18 × speedup in long-context generation;

(3) We conduct comprehensive experiments on tasks and models, which further demonstrate the superiority and robustness of our PolarQuant across a wide range of model families and tasks.

2 Background

Consider a specific Transformer layer where the input hidden states to the attention block are denoted as X ∈ R L × D , where L is the sequence length and D is the hidden state dimension. For any attention head, the d -dimensional query, key, and value vectors are obtained by applying three linear transformations to X . Specifically, for each head h , the corresponding computations are as follows:

where each W ∗ ∈ R D × d , and the resulting variables have shapes of R L × d .

The query and key vectors are then applied with RoPE [26] to incorporate positional information. For a query or key vector at position m ∈ [1 , L ] , the corresponding rotary matrix R m, Φ ∈ R d × d is defined as:

where O is a zero matrix, and each r m,ϕ i is a 2 × 2 orthogonal matrix. Here, ϕ i is typically defined as ϕ i = b -2 i/d , where b is the hyperparameter for RoPE base.

This formulation encodes the relative distance m -n between a query at position m>n and a key at position n into their inner product, as shown by:

the keys (after applying RoPE) and values, specifically ˜ K n R n, Φ (which we abbreviate as K n ) and V n , are thus cached to avoid re-computation, known as the KV cache. The KV cache leads to increased memory usage when processing long-context inputs.

Quantization is a simple but effective way to reduce the KV cache size. To clarify the mechanism, we provide a brief overview of key-value quantization below. For the value at position n , we follow the token-wise paradigm in [19] and quantize V n ∈ R d into b -bit, denoted as Q ( V n ) .

For an arbitrary dimension 0 ≤ j < d , we have:

where z n = min( V n [:]) is the zero-point, s n = (max( V n [:]) -min( V n [:])) / ( 2 b -1 ) is the scaling factor. The colon here denotes iteration over all dimensions, following Python indexing syntax. The function Clamp ( x, min, max ) restricts x to integers within the range [ min,max ] .

Outliers in key states make per-token quantization challenging, as we discussed earlier and illustrated in Figure 1(a). To address this, previous approaches [19, 13] quantize key vectors channel-wise.

For an arbitrary dimension j , a quantized key Q ( K n ) at position n is given by:

where the zero-point and scaling factor alternate as:

Here, the colon in the subscript denotes iteration over all token positions.

3 Method

Webegin by presenting the key findings of the activation patterns in the key cache (Section 3.1). These insights serve as the foundation for our proposed quantization approach, PolarQuant (Section 3.2).

3.1 Motivation

As discussed earlier, outliers in key vectors pose a dilemma for key cache quantization. Our solution to this challenge arises from a key observation:

Observation: When mapping the paired dimensions with outliers to polar coordinates, the resulting 2D vectors show consistent magnitudes. The outliers present in one channel are compensated by the activations of the other, which significantly simplifies quantization.

Recall that in key vectors, elements in certain dimensions are jointly rotated by the same rotary sub-matrix r n,ϕ i . Our analysis shows that the most prominent outliers tend to occur in only one of these dimension pairs. 5

Figure 1 (b) maps the paired dimensions from Figure 1 (a) onto a 2D Cartesian coordinate system, where the x-axis represents the first dimension and the y-axis represents the second.

Quantization

Range

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Channel-wiseMagnitudeofPairedDimension

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MagnitudeofMappedVectorson2DCartesianCoordinate

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Despite large variations in individual x and y values (which would indicate outliers in isolation), the mapped vectors form a well-structured pattern. In other words, when transformed into polar coordinates, the outliers are characterized by a smoothly distributed radial coordinate r and a polar angle θ . This structure significantly alleviates the quantization challenges faced by key caches.

Figure 2 provides a supplementary illustration of how this polar transformation improves quantization. The top figure illustrates a pair of dimensions exhibiting outliers, each with a large individual value range. Quantizing this large range inevitably results in a loss of precision. In the bottom figure, by combining these dimensions into a 2D vector, the norm (i.e., the polar radius) shows a significantly narrowed value range, facilitating quantization.

3.2 PolarQuant: Polar-coordinate-based quantization of post-RoPE key cache

Building on these insights, we introduce a novel key cache quantization approach based on polar transformation. Since the benefits of adopting a polar-coordinate have been outlined in the previous subsection, here we focus on the implementation details.

For an arbitrary 2-dimensional subvector ( K n [2 j ] , K n [2 j +1] ) in the key cache at position n , where 0 ≤ j < d/ 2 , we interpret ( K n [2 j ] , K n [2 j +1] ) as Cartesian coordinates in the xy -plane. This 2D vector is then converted to polar coordinates, where the radius ρ n [ j ] is given by:

and the polar angle is:

where atan2 ( y, x ) returns the angle between the positive x-axis and the point ( x, y ) .

We perform asymmetric group-wise quantization on ρ n [ j ] and θ n [ j ] , using a group size of g , with r -bit precision for ρ n [ j ] and t -bit precision for θ n [ j ] :

where z ρ [:] [ j ] , z θ [:] [ j ] are the zero-points and s ρ [:] [ j ] , s θ [:] [ j ] are the scaling factors:

Intuitively, PolarQuant partitions the two-dimensional plane into 2 r + t regions, spanned by 2 r radii and 2 t polar angles. Each 2D sub-vector of the key vector is then represented by the center of the region in which it resides. Figure 1 provides an illustration of the quantization process. The corresponding Cartesian coordinates in the key vector at dimensions 2 j and 2 j +1 are then calculated for the quantized representation ( Q ( ρ n [ j ]) , Q ( θ n [ j ]) ) , which is formulated as:

where ˜ ρ n [ j ] and ˜ θ n [ j ] are the dequantized radius and polar angle:

3.3 Efficient decoding with PolarQuant

In this section, we present PolarQuant's design for query-key multiplication, which incorporates the idea of post-multiplication dequantization to achieve acceleration of the decoding process. We begin by reviewing the conventional approach to dequantized generation. Specifically, during the generation phase, the cached keys must be dequantized before being multiplied by the current query Q m at position m . For each dimension 0 ≤ j < d , we have:

where ˜ K n denotes the dequantized key, and the inner product is then computed as Q m · ˜ K [:] .

The standard dequantization-then-multiplication operation introduces overhead for PolarQuant, as it demands extra computation. We argue that this overhead is redundant. At any dimension j , the dequantized outcomes come from a finite set of size 2 r + t . Since the cache size far exceeds this set, it is more efficient to pre-compute and reuse the post-multiplication intermediate results using a lookup table (LUT). This approach of leveraging LUTs has been explored in prior studies [13, 35]. KVQuant adopts an LUT to dequantize the key cache and restores positional information via RoPE recomputation. PolarQuant, in contrast, avoids RoPE recomputation by directly constructing a LUT for QK product on the fly. This is the key insight behind how PolarQuant accelerates decoding.

Specifically, PolarQuant builds its lookup table within each channel by mapping quantized polar coordinates to Cartesian coordinates and computing the dot products with the query sub-vectors. We implement custom Triton kernels to perform fused dequantization and query-key multiplication for efficient GPU execution of PolarQuant. The breakdown time analysis presented in Section 4.2 further demonstrates the effectiveness of our PolarQuant implementation. More implementation details can be found in Appendix A and our released code.

4 Experiment

In this section, we evaluate the performance of PolarQuant to highlight its effectiveness and efficiency . Our empirical results confirm that PolarQuant can be integrated with LLMs, while maintaining nearlossless performance of generative tasks (Section 4.1). We also highlight the speedup achieved by PolarQuant to showcase the superiority of our decoding algorithm and implementation (Section 4.2).

4.1 Preserving performance in quantized language and reasoning models

General setup. To ensure fair comparisons, we retain the value cache in full precision to avoid potential bias from its quantization in downstream tasks. We evaluate PolarQuant against several quantization baselines built on HuggingFace Transformers codebase [31]. For all group-wise quantization methods, the group size is fixed at g = 128 . Additional details about the baseline methods and experimental setup are provided in Appendix B. We evaluate all baselines alongside our PolarQuant on a range of models from mainstream model families, including Llama [1, 28] and Qwen [27].

Quantizing language models. In real-world applications of language models, the key-value cache often becomes the primary memory bottleneck when processing long-context inputs. We evaluate PolarQuant and baseline methods on LongBench [5], a widely used benchmark for long-context evaluation. Table 1 presents results on Qwen and Llama. There are two advantages of PolarQuant:

PolarQuant performs robustly across different model backbones. Quantization is especially challenging for Qwen models, which exhibit extreme channel-wise outliers in their key cache. 6 When applied to these models, token-wise quantization methods - such as pertoken Int. and ZipCache - tend to collapse. In contrast, KIVI-4 and PolarQuant constrain the accuracy drop to within 10%. For Llama-3.1-8B-Instruct, PolarQuant even improves average performance under 3-bit quantization, whereas all baseline methods, including KIVI, result in performance degradation.

Averaged across all evaluated settings-covering both model families and quantization precisions-PolarQuant achieves the best overall performance preservation.

Beyond long-context processing, we also apply PolarQuant on LLMs to examine their generative abilities with normal-length inputs. To explore how quantization affects the emergent capabilities of LLMs, such as chain-of-thought reasoning [30] and in-context learning [6], we benchmark PolarQuant's performance on the 5-shot CoT GSM8K [8]. The results, presented in Table 2, show that PolarQuant effectively supports both short- and long-context inputs, without compromising performance on reasoning or knowledge-intensive tasks.

Quantizing reasoning models. Large reasoning models (LRMs) exhibit remarkable capability in solving complex problems by long chains of thought. Recent studies [18] have shown that complex tasks-such as mathematics [16, 8] and reasoning [23]-are sensitive to the accumulation of quantization errors. Given that LRMs rely on both long-context processing and the generation of lengthy outputs, quantization techniques are particularly critical for their application. Moreover, the

underlying mechanisms of LRMs differ from those of LLMs, making quantization more challenging and offering valuable insights into the effectiveness of different approaches.

We apply key quantization to the distillation-based reasoning models of DeepSeek-R1 [20]. The accuracy scores across different quantization methods and tasks are presented in Table 3. The quantized reasoning models of PolarQuant achieve substantial improvements over the baselines, which provides strong supplementary evidence of PolarQuant's superiority.

4.2 Superior efficiency of PolarQuant

To evaluate the efficiency of the customized decoding algorithm, we provide a comprehensive time breakdown analysis of our PolarQuant implementation. In all experiments, we use the Llama-3.1-8BInstruct model configuration, which includes 32 query heads of dimension 128, and 8 key/value heads (grouped-query attention [2]). We benchmark the latency of our tailored kernel implementation, as well as the end-to-end generation throughput.

Latency for query-key multiplication kernel. We make comparisons of the query-key multiplication implementations for LLM decoding. Specifically, we evaluate the wall-clock latency across different sequence lengths and batch size settings. We benchmark the runtime by summing across 10K iterations of the calculations, and report results for: Fp16 , KIVI-4 , KIVI-2 , PolarQuant 44 and PolarQuant 33 . Figure 3 illustrates the performance comparison among kernel implementations, while Table 4 reports the multiplication latency with a batch size of 8. We see that, PolarQuant achieves up to 2.7 × speedup compared with KIVI and 1.6 × speedup compared with Fp16 Torch implementation.

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Throughput comparison. We incorporate our custom kernels into the generation pipeline and measure their end-to-end performance. We determine the maximum supported batch size with a sequence length of 32K tokens, with Hugging Face's implementation. We fix the input length at 256 tokens and measure throughput across different generation lengths. Table 4 presents the results; entries marked with † indicate the application of 2-bit value quantization. As is shown in Table 4, PolarQuant achieves up to 3.18 × throughput improvements, and demonstrates significant improvements over KIVI in both latency and throughput. Values in parentheses denote the speedup.

Configuration

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5 Discussions

5.1 Ablation studies of PolarQuant

Effect of group size g . PolarQuant applies group-wise quantization along the channel dimension. We conduct ablation studies to investigate the impact of the group size g on model performance.

Specifically, we benchmark PolarQuant and KIVI on LongBench using the Llama-3.1-8B-Instruct model. The results, presented in Table 5, indicate that PolarQuant performs competitively with KIVI across all tested group sizes. It is noteworthy that the quantization parameters occupy 32 /g bits; therefore, smaller values of g result in higher average bit widths. To strike an optimal balance between performance and parameter overhead, we set g as 128 throughout this paper.

Impact of RoPE configuration. We conduct experiments to test the sensitivity of PolarQuant to different RoPE configurations and draw the following conclusions:

(1) PolarQuant exhibits consistent performance across LLMs with different RoPE base frequencies (see Table 1). The experimental results for basic values of { 10000 , 500000 , 1000000 } , highlight PolarQuant's insensitivity to the choice of base frequency.

(2) PolarQuant is adaptable to different RoPE variants. We employ NTK RoPE scaling [24] to extend the LLM's context and apply PolarQuant to the key cache. Critically, mo significant performance drop is observed. We provide detailed experimental results and setups in Appendix C.

Bitwidth allocation for radii and polar angles. In PolarQuant, we assign bitwidth asymmetrically between radii and angles. To determine which component requires higher precision, we conduct ablation studies on various bitwidth configurations, enabling a more flexible allocation strategy.

We take Llama-3.1-8B-Instruct as backbone and benchmark PolarQuant on LongBench, to assess the impact of different bitwidth settings. From Table 6, we have the following observations:

Observation 1: Angle quantization is more sensitive to bitwidth. Allocating fewer than 3 bits to angles often results in a significant drop in performance.

Observation 2: Despite the asymmetry in bitwidth allocation, a more balanced distribution between radii and angles can still achieve strong performance.

5.2 Compatibility with existing KV cache compression techniques

In this section, we explore the integration of existing KV compression techniques with PolarQuant, which allows for further reductions in KV cache memory occupation. We present the experimental results of Llama-3.1-8B-Instruct on LongBench. As stated in Section 4.1, we retain the value cache in full precision, as key cache is more sensitive to low-precision quantization. Appendix D provides further analysis to support this. We combine value quantization with PolarQuant to verify its compatibility. A standard token-wise quantization is applied to the value cache, consistent with KIVI [19]. Table 7 presents the results. The introduction of value quantization results in only marginal performance degradation, even at 2-bit precision.

We further explore the integration of PolarQuant with token-eviction strategies [15]. As shown in Table 8, PolarQuant does not exhibit significant performance degradation. We left it for future work to combine PolarQuant with existing mixed-precision quantization for further memory saving [3, 10, 33].

6 Conclusion

In this paper, we view the outliers in the key cache of LLMs from a novel polar-coordinate-based perspective, which provides an efficient and effective solution, PolarQuant, to reduce the complexity and quantization costs in previous methods. PolarQuant well preserves downstream performance even in long-context understanding and long chain-of-thought generation, comparable to previous works under 4-bit precision while achieving superior efficiency. We hope the polar coordinate view can inspire the community to advance new low-bit precision quantization techniques.

Acknowledgments

This work is supported by Meituan through Agentic system X Program. Songhao Wu is supported by "Qiushi Academic-Dongliang" Project of Renmin University of China (No. RUC24QSDL015). We appreciate all reviewers for their valuable comments and suggestions, which are crucial for improving our work. We are also grateful to Xintong Qiu, Jixiang Hong, Tao Tan, Jia-Nan Li and Yuxuan Liu for their insightful suggestions on the manuscript.

References

AI@Meta. Llama 3 model card. 2024. URL https://github.com/meta-llama/llama3/ blob/main/MODEL_CARD.md .

Joshua Ainslie, James Lee-Thorp, Michiel de Jong, Yury Zemlyanskiy, Federico Lebron, and Sumit Sanghai. GQA: Training generalized multi-query transformer models from multi-head checkpoints. In Houda Bouamor, Juan Pino, and Kalika Bali, editors, Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing , pages 4895-4901, Singapore, December 2023. Association for Computational Linguistics. doi: 10.18653/v1/2023. emnlp-main.298. URL https://aclanthology.org/2023.emnlp-main.298/ .

Saleh Ashkboos, Amirkeivan Mohtashami, Maximilian L. Croci, Bo Li, Pashmina Cameron, Martin Jaggi, Dan Alistarh, Torsten Hoefler, and James Hensman. Quarot: Outlier-free 4-bit inference in rotated llms, 2024. URL https://arxiv.org/abs/2404.00456 .

Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate, 2016. URL https://arxiv.org/abs/1409.0473 .

Yushi Bai, Xin Lv, Jiajie Zhang, Hongchang Lyu, Jiankai Tang, Zhidian Huang, Zhengxiao Du, Xiao Liu, Aohan Zeng, Lei Hou, Yuxiao Dong, Jie Tang, and Juanzi Li. Longbench: A bilingual, multitask benchmark for long context understanding. arXiv preprint arXiv:2308.14508 , 2023.

Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. Language models are few-shot learners, 2020. URL https://arxiv.org/abs/2005.14165 .

Zefan Cai, Yichi Zhang, Bofei Gao, Yuliang Liu, Tianyu Liu, Keming Lu, Wayne Xiong, Yue Dong, Baobao Chang, Junjie Hu, and Wen Xiao. Pyramidkv: Dynamic kv cache compression based on pyramidal information funneling, 2024. URL https://arxiv.org/abs/2406. 02069 .

Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. Training verifiers to solve math word problems. arXiv preprint arXiv:2110.14168 , 2021.

DeepSeek-AI, Aixin Liu, Bei Feng, Bin Wang, Bingxuan Wang, Bo Liu, Chenggang Zhao, Chengqi Dengr, Chong Ruan, Damai Dai, Daya Guo, Dejian Yang, Deli Chen, Dongjie Ji, Erhang Li, Fangyun Lin, Fuli Luo, Guangbo Hao, Guanting Chen, Guowei Li, H. Zhang, Hanwei Xu, Hao Yang, Haowei Zhang, Honghui Ding, Huajian Xin, Huazuo Gao, Hui Li, Hui Qu, J. L. Cai, Jian Liang, Jianzhong Guo, Jiaqi Ni, Jiashi Li, Jin Chen, Jingyang Yuan, Junjie Qiu, Junxiao Song, Kai Dong, Kaige Gao, Kang Guan, Lean Wang, Lecong Zhang, Lei Xu, Leyi Xia, Liang Zhao, Liyue Zhang, Meng Li, Miaojun Wang, Mingchuan Zhang, Minghua Zhang, Minghui Tang, Mingming Li, Ning Tian, Panpan Huang, Peiyi Wang, Peng Zhang, Qihao Zhu, Qinyu Chen, Qiushi Du, R. J. Chen, R. L. Jin, Ruiqi Ge, Ruizhe Pan, Runxin Xu, Ruyi Chen, S. S. Li, Shanghao Lu, Shangyan Zhou, Shanhuang Chen, Shaoqing Wu, Shengfeng Ye, Shirong Ma, Shiyu Wang, Shuang Zhou, Shuiping Yu, Shunfeng Zhou, Size Zheng, T. Wang, Tian Pei, Tian Yuan, Tianyu Sun, W. L. Xiao, Wangding Zeng, Wei An,

Wen Liu, Wenfeng Liang, Wenjun Gao, Wentao Zhang, X. Q. Li, Xiangyue Jin, Xianzu Wang, Xiao Bi, Xiaodong Liu, Xiaohan Wang, Xiaojin Shen, Xiaokang Chen, Xiaosha Chen, Xiaotao Nie, Xiaowen Sun, Xiaoxiang Wang, Xin Liu, Xin Xie, Xingkai Yu, Xinnan Song, Xinyi Zhou, Xinyu Yang, Xuan Lu, Xuecheng Su, Y. Wu, Y. K. Li, Y. X. Wei, Y. X. Zhu, Yanhong Xu, Yanping Huang, Yao Li, Yao Zhao, Yaofeng Sun, Yaohui Li, Yaohui Wang, Yi Zheng, Yichao Zhang, Yiliang Xiong, Yilong Zhao, Ying He, Ying Tang, Yishi Piao, Yixin Dong, Yixuan Tan, Yiyuan Liu, Yongji Wang, Yongqiang Guo, Yuchen Zhu, Yuduan Wang, Yuheng Zou, Yukun Zha, Yunxian Ma, Yuting Yan, Yuxiang You, Yuxuan Liu, Z. Z. Ren, Zehui Ren, Zhangli Sha, Zhe Fu, Zhen Huang, Zhen Zhang, Zhenda Xie, Zhewen Hao, Zhihong Shao, Zhiniu Wen, Zhipeng Xu, Zhongyu Zhang, Zhuoshu Li, Zihan Wang, Zihui Gu, Zilin Li, and Ziwei Xie. Deepseek-v2: A strong, economical, and efficient mixture-of-experts language model, 2024. URL https://arxiv.org/abs/2405.04434 .

Shichen Dong, Wen Cheng, Jiayu Qin, and Wei Wang. Qaq: Quality adaptive quantization for llm kv cache, 2024. URL https://arxiv.org/abs/2403.04643 .

Nathan Habib, Clémentine Fourrier, Hynek Kydlíˇ cek, Thomas Wolf, and Lewis Tunstall. Lighteval: A lightweight framework for llm evaluation, 2023. URL https://github.com/ huggingface/lighteval .

Yefei He, Luoming Zhang, Weijia Wu, Jing Liu, Hong Zhou, and Bohan Zhuang. Zipcache: Accurate and efficient kv cache quantization with salient token identification, 2024. URL https://arxiv.org/abs/2405.14256 .

Coleman Hooper, Sehoon Kim, Hiva Mohammadzadeh, Michael W. Mahoney, Yakun Sophia Shao, Kurt Keutzer, and Amir Gholami. Kvquant: Towards 10 million context length llm inference with kv cache quantization, 2024. URL https://arxiv.org/abs/2401.18079 .

Hao Kang, Qingru Zhang, Souvik Kundu, Geonhwa Jeong, Zaoxing Liu, Tushar Krishna, and Tuo Zhao. Gear: An efficient kv cache compression recipe for near-lossless generative inference of llm, 2024. URL https://arxiv.org/abs/2403.05527 .

Yuhong Li, Yingbing Huang, Bowen Yang, Bharat Venkitesh, Acyr Locatelli, Hanchen Ye, Tianle Cai, Patrick Lewis, and Deming Chen. Snapkv: Llm knows what you are looking for before generation. arXiv preprint arXiv:2404.14469 , 2024.

Hunter Lightman, Vineet Kosaraju, Yura Burda, Harri Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe. Let's verify step by step, 2023. URL https://arxiv.org/abs/2305.20050 .

Nelson F. Liu, Kevin Lin, John Hewitt, Ashwin Paranjape, Michele Bevilacqua, Fabio Petroni, and Percy Liang. Lost in the middle: How language models use long contexts. Transactions of the Association for Computational Linguistics , 12:157-173, 2024. doi: 10.1162/tacl_a_00638. URL https://aclanthology.org/2024.tacl-1.9/ .

Ruikang Liu, Yuxuan Sun, Manyi Zhang, Haoli Bai, Xianzhi Yu, Tiezheng Yu, Chun Yuan, and Lu Hou. Quantization hurts reasoning? an empirical study on quantized reasoning models, 2025. URL https://arxiv.org/abs/2504.04823 .

Zirui Liu, Jiayi Yuan, Hongye Jin, Shaochen (Henry) Zhong, Zhaozhuo Xu, Vladimir Braverman, Beidi Chen, and Xia Hu. Kivi: a tuning-free asymmetric 2bit quantization for kv cache. In Proceedings of the 41st International Conference on Machine Learning , ICML'24. JMLR.org, 2025.

Liangchen Luo, Yinxiao Liu, Rosanne Liu, Samrat Phatale, Meiqi Guo, Harsh Lara, Yunxuan Li, Lei Shu, Yun Zhu, Lei Meng, Jiao Sun, and Abhinav Rastogi. Improve mathematical reasoning in language models by automated process supervision, 2024. URL https://arxiv.org/abs/ 2406.06592 .

OpenAI. Learning to reason with llms. https://openai.com/index/ learning-to-reason-with-llms/ , 2024. [Accessed 19-09-2024].

Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Köpf, Edward Yang, Zach DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, high-performance deep learning library, 2019. URL https://arxiv.org/abs/1912.01703 .

David Rein, Betty Li Hou, Asa Cooper Stickland, Jackson Petty, Richard Yuanzhe Pang, Julien Dirani, Julian Michael, and Samuel R. Bowman. Gpqa: A graduate-level google-proof q&a benchmark, 2023. URL https://arxiv.org/abs/2311.12022 .

Baptiste Rozière, Jonas Gehring, Fabian Gloeckle, Sten Sootla, Itai Gat, Xiaoqing Ellen Tan, Yossi Adi, Jingyu Liu, Romain Sauvestre, Tal Remez, Jérémy Rapin, Artyom Kozhevnikov, Ivan Evtimov, Joanna Bitton, Manish Bhatt, Cristian Canton Ferrer, Aaron Grattafiori, Wenhan Xiong, Alexandre Défossez, Jade Copet, Faisal Azhar, Hugo Touvron, Louis Martin, Nicolas Usunier, Thomas Scialom, and Gabriel Synnaeve. Code llama: Open foundation models for code, 2024. URL https://arxiv.org/abs/2308.12950 .

Noam Shazeer. Fast transformer decoding: One write-head is all you need, 2019. URL https://arxiv.org/abs/1911.02150 .

Jianlin Su, Yu Lu, Shengfeng Pan, Ahmed Murtadha, Bo Wen, and Yunfeng Liu. Roformer: Enhanced transformer with rotary position embedding, 2023. URL https://arxiv.org/ abs/2104.09864 .

Qwen Team. Qwen2.5: A party of foundation models, September 2024. URL https:// qwenlm.github.io/blog/qwen2.5/ .

Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, Dan Bikel, Lukas Blecher, Cristian Canton Ferrer, Moya Chen, Guillem Cucurull, David Esiobu, Jude Fernandes, Jeremy Fu, Wenyin Fu, Brian Fuller, Cynthia Gao, Vedanuj Goswami, Naman Goyal, Anthony Hartshorn, Saghar Hosseini, Rui Hou, Hakan Inan, Marcin Kardas, Viktor Kerkez, Madian Khabsa, Isabel Kloumann, Artem Korenev, Punit Singh Koura, Marie-Anne Lachaux, Thibaut Lavril, Jenya Lee, Diana Liskovich, Yinghai Lu, Yuning Mao, Xavier Martinet, Todor Mihaylov, Pushkar Mishra, Igor Molybog, Yixin Nie, Andrew Poulton, Jeremy Reizenstein, Rashi Rungta, Kalyan Saladi, Alan Schelten, Ruan Silva, Eric Michael Smith, Ranjan Subramanian, Xiaoqing Ellen Tan, Binh Tang, Ross Taylor, Adina Williams, Jian Xiang Kuan, Puxin Xu, Zheng Yan, Iliyan Zarov, Yuchen Zhang, Angela Fan, Melanie Kambadur, Sharan Narang, Aurelien Rodriguez, Robert Stojnic, Sergey Edunov, and Thomas Scialom. Llama 2: Open foundation and fine-tuned chat models, 2023. URL https://arxiv.org/abs/2307.09288 .

Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Ł ukasz Kaiser, and Illia Polosukhin. Attention is all you need. In I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems , volume 30. Curran Associates, Inc., 2017. URL https://proceedings.neurips.cc/paper_files/paper/2017/file/ 3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf .

Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Brian Ichter, Fei Xia, Ed Chi, Quoc Le, and Denny Zhou. Chain-of-thought prompting elicits reasoning in large language models, 2023. URL https://arxiv.org/abs/2201.11903 .

Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander M. Rush. Transformers: State-of-theart natural language processing. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations , pages 38-45, Online, October 2020. Association for Computational Linguistics. URL https://www.aclweb.org/anthology/ 2020.emnlp-demos.6 .

An Yang, Anfeng Li, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chang Gao, Chengen Huang, Chenxu Lv, Chujie Zheng, Dayiheng Liu, Fan Zhou, Fei Huang, Feng Hu, Hao Ge, Haoran Wei, Huan Lin, Jialong Tang, Jian Yang, Jianhong Tu, Jianwei Zhang, Jianxin Yang, Jiaxi Yang, Jing Zhou, Jingren Zhou, Junyang Lin, Kai Dang, Keqin Bao, Kexin Yang, Le Yu, Lianghao Deng, Mei Li, Mingfeng Xue, Mingze Li, Pei Zhang, Peng Wang, Qin Zhu, Rui Men, Ruize Gao, Shixuan Liu, Shuang Luo, Tianhao Li, Tianyi Tang, Wenbiao Yin, Xingzhang Ren, Xinyu Wang, Xinyu Zhang, Xuancheng Ren, Yang Fan, Yang Su, Yichang Zhang, Yinger Zhang, Yu Wan, Yuqiong Liu, Zekun Wang, Zeyu Cui, Zhenru Zhang, Zhipeng Zhou, and Zihan Qiu. Qwen3 technical report, 2025. URL https://arxiv.org/abs/2505.09388 .

June Yong Yang, Byeongwook Kim, Jeongin Bae, Beomseok Kwon, Gunho Park, Eunho Yang, Se Jung Kwon, and Dongsoo Lee. No token left behind: Reliable kv cache compression via importance-aware mixed precision quantization, 2024. URL https://arxiv.org/abs/2402. 18096 .

Amir Zandieh, Majid Daliri, and Insu Han. Qjl: 1-bit quantized jl transform for kv cache quantization with zero overhead, 2024. URL https://arxiv.org/abs/2406.03482 .

Hailin Zhang, Xiaodong Ji, Yilin Chen, Fangcheng Fu, Xupeng Miao, Xiaonan Nie, Weipeng Chen, and Bin Cui. Pqcache: Product quantization-based kvcache for long context llm inference. Proceedings of the ACM on Management of Data , 3(3):1-30, 2025.

Zhenyu Zhang, Ying Sheng, Tianyi Zhou, Tianlong Chen, Lianmin Zheng, Ruisi Cai, Zhao Song, Yuandong Tian, Christopher Ré, Clark Barrett, Zhangyang Wang, and Beidi Chen. H 2 o: Heavy-hitter oracle for efficient generative inference of large language models, 2023. URL https://arxiv.org/abs/2306.14048 .

Yilong Zhao, Chien-Yu Lin, Kan Zhu, Zihao Ye, Lequn Chen, Size Zheng, Luis Ceze, Arvind Krishnamurthy, Tianqi Chen, and Baris Kasikci. Atom: Low-bit quantization for efficient and accurate llm serving. In P. Gibbons, G. Pekhimenko, and C. De Sa, editors, Proceedings of Machine Learning and Systems , volume 6, pages 196209, 2024. URL https://proceedings.mlsys.org/paper_files/paper/2024/file/ 5edb57c05c81d04beb716ef1d542fe9e-Paper-Conference.pdf .

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A Accelerated Query-Key Inner Product with Cached Keys in PolarQuant

For a quantized representation ( Q ( ρ n [ j ]) , Q ( θ n [ j ]) ) , we first dequantize the polar coordinates back into Cartesian coordinates, which serves as the at dimensions 2 j and 2 j +1 :

where ˜ ρ n [ j ] , ˜ θ n [ j ] is the dequantization for ( ρ, θ ) respectively. ˜ ρ n [ j ] and ˜ θ n [ j ]) are formulated as:

( Q ( ρ n [ j ]) , Q ( θ n [ j ]) ) represents [ ˜ K n [2 j ] , ˜ K n [2 j +1] ] as a state in the lookup table.

When computing the query-key inner product for the dimensions 2 j and 2 j +1 , the result is:

and the final inner product is the sum:

Figure 4 presents a naive PyTorch [22] implementation of the aforementioned dequantization-andmultiplication operation. We implement a fused Triton kernel that reproduces the functionality of the PyTorch code for improved computational efficiency. More details can be found in our code.

B Details experiment setup

In this section, we provide additional details about the experimental setups.

B.1 Setup of baseline methods

This section provides an overview of the baseline methods. Following this, we outline the quantization configurations and variants for both the baselines and our proposed PolarQuant. We also explain how the average number of bits for the quantization parameters is calculated.

IntN applies token-wise N-bit quantization to the key states. This token-wise quantization incurs 32 /d bits quantization parameters (16 bits for zero-points and 16 bits for scaling factors per token), which amounts to 0.25 bits per token when d = 128 .

ZipCache -N [12] introduces a channel-separable, token-wise scheme for key quantization, where N denotes the quantization bits. Each key channel is normalized by the square root of its maximum magnitude before quantization. Similar to Int -N , this method performs token-wise quantization, allocating 0.25 bits for zero-points and scales.

KIVI -N [19] employs an asymmetric strategy for KV cache quantization, applying channel-wise quantization to the key cache and token-wise quantization to the value cache. For 4-bit quantization, we use KIVI -4 with a group size of 128. For 3-bit quantization, we use KIVI -2 with a group size of 32, as the official implementation does not support 3-bit quantization. This channel-wise quantization introduces (16+16) d bits of quantization parameters per group, which increases the average bitwidth by 32 /g : 1 bit for g = 32 and 0.25 bits for g = 128 respectively.

QJL [34] applies Johnson-Lindenstrauss transformation to key states, removing the memory overheads associated with storing quantization constants. For a 3-bit quantization schema, this method achieves a key cache bitwidth of 3.13 bits.

PolarQuant rt assigns r bits for radii quantization and t bits for polar angles, resulting in ( r + t ) / 2 bitwidth for key states quantization. PolarQuant also employs channel-wise quantization with grouping; this group partitioning adds an overhead of 32 /g bits.

Nearly all methods discussed here require a buffer size or residual length for applying quantization. We exclude the contribution of these residual key states to bit counts for comparisons.

B.2 Configurations of the open-sourced LLMs

Qwen-2.5-1.5B-Instruct [27] is the instruction-tuned 1.5B Qwen2.5 model, which supports a context length of up to 131,072 tokens and features a base RoPE frequency of 1,000,000.

Llama-2-7B-Chat [28] has a context length of 4096 and base RoPE frequency of 10,000.

Llama-3.1-8B-Instruct [1] is an 8 billion parameter language model, designed to handle a context length of up to 131,072 tokens, and has the base RoPE frequency set to 500,000.

B.3 Reasoning model evaluation

For fair comparison, the evaluation code is built on the Huggingface Lighteval framework [11]. We use the default generation configuration for datasets splitting, and the EM score is reported in Table 3. More implementation details can be found in our released code.

C NTK RoPE scaling experiment

We adopt NTK RoPE scaling [24] to extend the context window of the Llama-2-7B-Chat model from 4096 to 8192 . Specifically, we implement dynamic RoPE updates based on the Hugging Face codebase. PolarQuantachieve an average performance of 32.44 on LongBench. Compared to the 32.15 score of the Bf16 baseline, PolarQuant's performance remains competitive.

D Sensitivity Analysis of Key-Value Quantization

In Section 5.2, we combine 4-bit key quantization PolarQuant 44 with 2-bit value quantization and observe minimal performance degradation. We further evaluate value quantization (KIVI, group size 128) on LongBench. By retaining the key cache at full precision, KIVI result in no performance drop. However, when the key is quantized to the same bitwidth while the value is kept at full precision, the performance drops significantly. Table 9 presents the results, the notation ( K b , V c ) denotes key quantization to b -bit and value quantization to c -bit.

E Limitation

Although PolarQuant achieves promising results in reducing storage and computational resources, we also discuss the limitations of our current work. This work focuses exclusively on decoder-only Transformer-based large language models (LLMs) that utilize rotary position embedding (RoPE) as the underlying position encoding mechanism. RoPE has become a prevalent choice in many state-of-the-art open-source LLMs. However, the effectiveness of PolarQuant when applied to models with alternative position encoding methods or attention mechanisms remains an open question and warrants further investigation. Furthermore, exploring more recent LLM backbones [32] is necessary, but due to time and computational limitations, we leave this part of the work for future research.

Peer reviews:
comments: SUMMARY:
The paper presents PolarQuant, an approach that applies a polar transform on the KV caches and quantizes the angle and radius.

STRENGTHS AND WEAKNESSES:
Strengths

- KV-cache compression is an important area for LLM efficiency (especially for long-context inference) and the intuition of using a polar transformation is very clever. I found the idea (and observations) interesting
- The authors provide a real latency evaluation and compare them against the SoTA KV-cache quantization schemes
- Paper is well-written

Weaknesses

- Figures are a bit complex and hard to understand

QUESTIONS:
1. Can you please explain how the idea can be mixed with rotation-based quantization techniques like [1]? For example, can't we apply Hadamards of size 2 and fuse it into the weights of the previous layers (for W_k and W_v) and then quantize the KV?

2. I found the figures a bit complex and not easy to understand. Can you please polish them and make them easier?

3. Can you provide an ablation on the group sizes during the quantization?

4. I would like to see a breakdown over the runtimes to see the overhead of getting polar coordinates (like atan2 operation) especially during the generation phase which is the main target of KV-cache quantization.

5. Can you please provide more details about the GPU architecture you have yoused for the benchmarking experiments in the paper? 

6. Do you have any explanation about the "Observation 1" in line 252? Why the angles need more precision here? Do you have any intuition?

[1] https://arxiv.org/abs/2404.00456

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-22

comments: SUMMARY:
This paper proposes to quantize KV cache through effectively addressing outlier challenges via polar transformation.
Specifically, the authors observe that although the individual axis may contain outliers, they collectively form stable circular patterns, making quantization of the original outliers easier.
Experimental results include models like Llama and Qwen, showing competitive performance with 3-4 bit quantization and up to 3.18x speedup.

STRENGTHS AND WEAKNESSES:
Strengths:
1. Polar transformation novelly exploits RoPE-induced circular patterns (Fig 1b) to mitigate channel-wise outliers.
2. Polar transformation enables decoding acceleration through lookup tables, speeding up attention computation.
3. Comprehensive evaluation on long-context (LongBench), normal context (GSM8K) and reasoning (DeepSeek-R1) across multiple model families.
4. Clear methodology and abundant implementation details.

Weaknesses:
1. Figure 1a claims to show "significantly larger activations" but lacks visual evidence of outliers (no obvious outliers in channel dimension).
2. Given that key value quantization aims to tackle memory bottleneck, the lookup table in PolarQuant incurs extra memory overhead, so it is unclear why memory usage of PolarQuant is smaller than KIVI quant (shown in Table 4).

QUESTIONS:
1. Why does Table 4 show PolarQuant using less memory than KIVI despite lookup tables?
2. How does group size $g$ (Sec 3.2) affect performance? With small $g$, there might not be outliers in each group. Will PolarQuant still maintain its superiority under that circumstance?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-30

comments: SUMMARY:
This paper introduces PolarQuant, a novel quantization method for the K-cache in large language models. The method leverages a polar coordinate transformation to address the challenge of outliers in key vectors, which traditionally hinder effective quantization. By encoding key vectors as quantized radii and angles, PolarQuant achieves both memory efficiency and decoding acceleration, while maintaining high performance across various tasks and model architectures.

STRENGTHS AND WEAKNESSES:
Strength:

- The use of polar coordinates to mitigate outlier effects in key vectors is a novel and elegant idea. The authors show that outliers often appear in only one of the two dimensions rotated by RoPE, and that these pairs form smooth circular patterns in polar space, making them more amenable to quantization.
- The authors provide detailed implementation and reproducibility information, which is commendable.

QUESTIONS:
- Recent models like Qwen3 have adopted a pre-RoPE normalization layer (https://github.com/huggingface/transformers/blob/main/src/transformers/models/qwen3/modeling_qwen3.py#L185), which retains the polar characteristics of RoPE while reducing the quantization burden. Would PolarQuant still be applicable or effective for such models?

- How was the value cache (V-cache) quantization method chosen? Was it selected because KIVI demonstrated the best performance among existing approaches

- Given that PolarQuant relies on the structured circular patterns induced by RoPE, how sensitive is the method to variations in RoPE configurations (e.g., different base frequencies or dimensional pairings)? Could this affect its generalizability across diverse model architectures or custom RoPE implementations?

LIMITATIONS:
-
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nPolarQuant aims to accurately quantize the key cache to low precision by leveraging a polar representation for the keys. Their approach is designed to address the impact of the RoPE embedding (which rotates neighboring channels in key activations). They observe that when representing pairs of channels in polar form, the outlier channels in keys (which are typically coupled with a non-outlier channel) become easier to quantize since they have a consistent norm along the sequence length dimension. By quantizing keys in polar form, they achieve superior efficiency (in terms of improved accuracy for the same bit width) relative to existing methods. They also provide an efficient system implementation that exploits table lookups to accelerate inference, showing the efficiency gains of their approach through both kernel-level and end-to-end benchmarking.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- They propose a novel polar transformation to tackle the effects of RoPE on the key distributions (which counteracts the effect of the rotation on widening the range of outlier channels).\n- They present convincing evaluation across multiple models (llama3.1 and qwen2.5) and datasets (LongBench + GSM8K) as well as comparisons with existing strong baselines like KIVI/QJL.\n- They present a lookup table mapping for efficient decoding as well as custom Triton kernels which show substantial latency gains with their method. Their speedup evaluation includes kernel-level and end-to-end evaluation.\n\nWeaknesses:\n- The insight that the key cache has outlier channels with consistent magnitude pre-rope and that the magnitude gets rotated with the neighboring channel (leading to varying magnitude along the sequence length) is not novel relative to prior work. Although they argue that their method is superior to applying RoPE during inference as the key cache is loaded (which is an alternative strategy to account for the rotation), they do not justify that the overhead of performing recomputation using RoPE is substantial (particularly since KV cache loading is memory bandwidth bound, this overhead may potentially be hidden / overlapped with memory loads).\n- The fast decoding method using lookup tables makes sense, but is not entirely novel in the context of KV cache quantization - for example, previous non-uniform quantization methods like KVQuant have leveraged lookup tables for fast dequantization. \n- Their method focuses mainly on efficient key quantization; however, even if we can push keys to low bit precision, we will still be limited by the precision with which we can represent values. A more detailed analysis of challenges with value quantization is not presented. Alternatively, some ablations demonstrating that the relative difficulty of value quantization relative to key quantization is low would be useful.\n\nQUESTIONS:\n- What is the overhead of quantizing new keys online during inference using this method - is it negligible?\n- Do you have any analysis demonstrating the relative higher sensitivity of the key cache to low-precision quantization (compared with values)? Also, is the key cache still more sensitive than the value cache even after applying the polar quant method?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}