Block-Clustered Quantization (BCQ)

• Block-Clustered Quantization (BCQ) is a method that partitions network parameters into blocks for joint quantization of weights and activations, ensuring minimal accuracy loss at low bit-widths.
• It employs adaptive scaling techniques like per-channel and per-block quantization to minimize errors and maintain performance in ultra-low precision regimes.
• BCQ enhances hardware efficiency by reducing memory footprint and power consumption, making it ideal for deployment on edge devices and specialized accelerators.

Block-Clustered Quantization (BCQ) refers to a family of neural network compression strategies that unify and optimize quantization across both weights and activations, often at fine granularity such as per-channel or per-block, to balance accuracy and hardware efficiency under aggressive reduction of bit-widths. While the terminology “block-clustered quantization” itself is not standardized in the cited literature, the underlying principle aligns closely with activation-quantization-aware scaling (AQAS) and harmonized quantization methods employed in contemporary LLMs and image models. These approaches orchestrate joint or adaptive quantization operations such that information loss is minimized even at sub-4-bit precision, a regime where naive independent quantization of weights and activations typically leads to severe performance collapse. This article surveys the conceptual foundations, mathematical formulations, empirical results, hardware consequences, and practical guidance for block-clustered and AQAS-inspired quantization strategies as documented in recent arXiv research.

1. Conceptual Foundations and Motivations

Block-clustered quantization schemes aim to compress neural networks by partitioning parameters (weights and/or activations) into blocks, clusters, or channels and then quantizing each group with either shared or individually optimized scaling factors and bit-widths. The motivation arises from the observation that the precision requirements and distributional properties of weights and activations can differ substantially across layers, channels, or clusters. Independent quantization often induces severe bias or outlier-induced errors in low-bit regimes (1–4 bits), notably in LLMs where weights are tightly clustered but activations may have heavy-tailed distributions, or vice versa (Lee et al., 2023).

AQAS and harmonized post-training quantization methods instantiate the block-clustered spirit by explicitly optimizing quantization parameters to jointly minimize end-to-end loss or error, rather than treating each tensor independently. The interaction between weight-induced structural changes and activation-driven pixel or token-level fidelity is central, dictating the accuracy-memory-computation trade-offs at low bit-widths (Wang et al., 8 Nov 2025).

2. Mathematical Formulation and Scaling Laws

A unified scaling law framework underpins block-clustered quantization, allowing principled comparison and combination of sparsity and quantization effects. Given a neural model with $N$ raw parameters, the effective parameter count under quantization is

$N_{\text{eff}} = N \cdot \mathrm{eff}(q_w, q_a)$

where $q_w$ and $q_a$ denote the bit-widths for weights and activations, and $\mathrm{eff}(\cdot, \cdot)$ is the effective parameter multiplier (EPM) empirically fitted for each quantization regime (Frantar et al., 23 Feb 2025). This multiplier reflects the mean information rate per compressed parameter relative to the full-precision (e.g. bfloat16) baseline.

All compressed model families empirically obey the same three-term scaling law for cross-entropy loss:

$$L(N, D, C) = \frac{a}{(N \cdot \mathrm{eff}(C))^b} + \frac{c}{D^d} + e$$

where $L$ is the training loss, $D$ is the token count, and $C$ captures the quantization configuration. Specialized forms distinguish between weight-only ($\mathrm{eff}_w(q_w)$) and full quantization of both weights and activations ($\mathrm{eff}_\text{full}(q)$).

Empirically, weight-only quantization achieves high EPMs at moderate bit-widths (e.g., $\mathrm{eff}_w(4) = 0.923$, $\mathrm{eff}_w(2) = 0.702$, $\mathrm{eff}_w(1) = 0.466$), enabling substantial parameter savings (Frantar et al., 23 Feb 2025). Full quantization of weights and activations shows diminishing returns below 4 bits ($\mathrm{eff}_\text{full}(4) = 0.747$, $\mathrm{eff}_\text{full}(2) = 0.289$, $\mathrm{eff}_\text{full}(1) = 0.067$), marking a regime where clustered or harmonized quantization strategies become essential.

3. Block-/Channel-Wise and Adaptive Quantization Strategies

Block-clustered quantization generally implements quantizer parameters (scales, zero-points) at channel, block, or group granularity, rather than at the entire tensor or layer level. Notable instantiations include:

• Vectorized per-channel scaling: GranQ replaces global scales with channel-wise or block-wise scales computed in parallel, minimizing quantization distortion and enabling fine-grained control (see GranQ pseudocode in (Hong et al., 24 Mar 2025)).
• Adaptive Step Size Quantization (ASQ): Incorporates lightweight neural modules to produce adaptive multipliers for activation scales, dynamically matching quantization resolution to the instantaneous distribution of each block or channel. This approach reduces both clipping and dead zones, particularly beneficial at ultra-low bit-widths (Zhou et al., 24 Apr 2025).
• Harmonized Scale Optimization: In HarmoQ, the optimal block scale $s^*$ equalizes weight and activation quantization errors, derived in closed form as

$$s^* = \sqrt{\frac{(\beta_x - \alpha_x)(2^{b_w}-1)}{(\beta_w - \alpha_w)(2^{b_x}-1)}}$$

where $[\alpha_x, \beta_x]$ and $[\alpha_w, \beta_w]$ are block-specific clipping bounds for activations and weights (Wang et al., 8 Nov 2025).

Block-clustered quantization can also leverage residual decomposition for binary activation quantization, learning smooth scaling factors per bit-plane within clusters to drive down quantization error (Song et al., 7 Apr 2025).

4. Empirical Results and Performance Impact

Block-clustered and AQAS-based quantization schemes exhibit consistent empirical gains across multiple domains:

Method	Bit-width	Accuracy/Metric	Relative to Baseline	arXiv ID
GranQ (ResNet-20, CIFAR-100)	3w3a	62.73% (classification)	+5.45% over GenQ, SOTA	(Hong et al., 24 Mar 2025)
ASQ+POST (ResNet-34, ImageNet)	4-bit	74.1% (Top-1)	+0.8% over FP, +1.9% over LSQ	(Zhou et al., 24 Apr 2025)
HarmoQ (SwinIR-Light, Set5)	2-bit	PSNR=36.45dB, SSIM=0.9482	+1.33dB, +0.0095 over baseline	(Wang et al., 8 Nov 2025)
SASQ (LLaMA2-7B, WikiText2)	8-bit A	PPL=4.651	–4.7% vs FP16, –5.2% vs QuaRot	(Mao et al., 16 Dec 2025)
W(1+1)A(1*4)+AQAS (LLaMA-1 7B)	binarized	PPL=8.58, QA Acc=54.9%	Near FP16 at 1–2 bit, +18% accuracy	(Song et al., 7 Apr 2025)

The key significance is that vectorized, adaptive, or harmonized clustering of quantization parameters across blocks, channels, or bit-planes enables maintenance of high accuracy and fidelity even at severely reduced bit-widths, outperforming global or static scaling methods.

5. Hardware Efficiency and Deployment Implications

Block-Clustered Quantization directly influences the computational and memory footprint of deployed models. Fine-grained scaling enables aggressive reduction of bit-widths (e.g., 1–4 bits for weights and activations), yielding substantial gains in memory and energy consumption. For instance, the dINT4×INT8 MAC unit synthesized for AQAS (Lee et al., 2023) achieves 1.93× area savings and 2.56× power savings over traditional INT8×INT8 MAC, while restoring accuracy to near full-precision.

Hardware efficiency is further enhanced by:

• Lookup Table (LUT) implementations for non-uniform quantization schemes (Zhou et al., 24 Apr 2025).
• Vectorized quantization kernels optimizing parallel scale application (Hong et al., 24 Mar 2025).
• Static per-channel scale storage (SASQ) with negligible overhead, suitable for edge deployment (Mao et al., 16 Dec 2025).

A plausible implication is that block-clustered quantization can be tailored to specific hardware architectures (e.g., NPUs, custom ASICs) to maximize throughput and memory efficiency while maintaining model quality.

6. Best Practices and Practical Recommendations

Recent research converges on several practical recommendations for implementing block-clustered or AQAS-driven quantization:

• Weight-only quantization delivers excellent parameter efficiency even at 1–2 bits, recommended when memory is the principal constraint (Frantar et al., 23 Feb 2025).
• Full weight+activation quantization finds its optimal regime at 4 bits, with substantial accuracy drop below this threshold. Clustering scales per block or adaptively is essential to preserve performance in ultra-low-bit scenarios (Frantar et al., 23 Feb 2025, Song et al., 7 Apr 2025).
• Mixed quantization (e.g., 4W2A) provides an accuracy-memory trade-off almost indistinguishable from symmetric bit-width regimes, writable as block-clustered (Frantar et al., 23 Feb 2025).
• Adaptive or harmonized scale learning via small neural modules or closed-form error balancing (Zhou et al., 24 Apr 2025, Wang et al., 8 Nov 2025), is effective for dealing with runtime distribution shifts, heavy-tailed activation blocks, or structural-image distortion asymmetry.
• Static per-block/channel scale optimization as in SASQ (Mao et al., 16 Dec 2025) supports deployment on edge devices with minimal training overhead and inference latency.

7. Limitations and Future Directions

Limitations of block-clustered quantization arise from the dependence on block-specific distribution statistics and the need for high-quality synthetic or calibration data to expose activation and weight range. Scale vectors may fluctuate if input data exhibits wide variability, and kernel-level vectorization must be efficiently mapped to hardware. Future research directions include:

• Joint optimization of scale parameters with regularization for mixed-precision or multi-block contexts (Hong et al., 24 Mar 2025).
• Development of hardware primitives explicitly supporting per-block/cluster scaling and dynamic update (Lee et al., 2023).
• Further exploration of harmonized quantization in domains exhibiting severe pixel-structural error asymmetry (Wang et al., 8 Nov 2025).

Block-clustered quantization and its AQAS variants thus provide a mathematically principled and empirically validated pathway to high-fidelity, hardware-efficient neural compression, unifying sparsity and quantization under scalable, adaptive parameter frameworks.