Asymmetric Quantization Scheme

• Asymmetric quantization scheme is a framework that maps real-valued signals to discrete values by adapting quantizer parameters to non-symmetric data distributions.
• It optimizes applications such as model compression and hardware acceleration by employing adaptive scale, offset, and non-uniform quantization strategies.
• The method finds use in analog-to-digital conversion, neural network quantization, and maximizing channel capacity in low-SNR communication systems.

An asymmetric quantization scheme is a statistical and algorithmic framework for mapping a set of real-valued data (such as signals, model weights, activations, or features) to a finite discrete set, in which the quantizer parameters (such as scaling, offset, or range) are not constrained to be symmetric about zero or a fixed reference, but are instead adapted to the typically non-symmetric distribution of the data. Asymmetric quantization has become a core principle in a variety of domains—ranging from analog-to-digital conversion to large-scale machine learning model compression—because most real-world data distributions are significantly skewed or biased toward one side, and because asymmetric error propagation and application-specific requirements often demand non-uniform treatment of quantization domains.

1. Core Concepts and Mathematical Formulations

The essential motivation for asymmetric quantization is that the empirical or statistical distribution of data (weights, signals, activations) is frequently not symmetric around zero. The general form of an asymmetric uniform quantizer is defined by

$$\bar{x} = Q(x, s, z, k) = \mathrm{clip}\left(\left\lfloor\frac{x}{s}\right\rceil - \lfloor z \rceil, 0, k\right),\qquad \hat{x} = s(\bar{x} + \lfloor z \rceil)$$

where:

• $s$ is a learned or estimated scaling factor (step size),
• $z$ is a real-valued offset or zero-point,
• $k = 2^b - 1$ is the range for a given bit-width $b$.

Parameterizations for $(s, z)$ include (i) direct learning of $s$ and $z$ ("scale/offset"), (ii) minimum ($\theta_\text{min}$) and maximum ($\theta_\text{max}$) bounds, and (iii) scaling factors ($\beta, \gamma$) allowing parametric modulation of the range: $$s = \frac{(\gamma)\theta_\text{max} - (\beta)\theta_\text{min}}{k},\qquad z = \frac{(\beta)\theta_\text{min}}{s}$$ Each parameterization scheme offers different stability and convergence properties during quantization-aware training (QAT) and post-training quantization (You et al., 25 Apr 2024).

In floating-point schemes, the asymmetry is implemented by allocating distinct scales to positive and negative sub-domains: $$\text{scale}_{\mathrm{pos}} = \frac{\max(w)}{\text{range}/2},~\text{scale}_{\mathrm{neg}} = \frac{-\min(w)}{\text{range}/2}$$ This allows quantization levels to separately track the range on each side of zero, maintaining maximal dynamic range and precision in the presence of skewed weight distributions (Zhang et al., 2023).

2. Asymmetric Quantization in Analog and Digital Signal Processing

In classic analog-to-digital conversion, asymmetric quantization was introduced to address the limitations of symmetric sigma-delta ($\Sigma\Delta$) schemes. In conventional second-order $\Sigma\Delta$ quantizers, the recursive update is symmetric, leading to persistent periodic cycles (idle tones) when the input signal vanishes: $$u_n = u_{n-1} + f_n - q_n,~v_n = u_{n-1} + v_{n-1} + f_n - q_n$$ The asymmetric modification introduces damping ($\rho < 1$) selectively when the state $u_n \ge 0$ (i.e., $M x_n := T(\rho x_n)$ if $d \cdot x_n \ge 0$), guaranteeing global convergence of the state $(u_n, v_n)$ to the origin under vanishing input. The proof employs an invariant (trapping) set and a Lyapunov-type function $V(u,v) = u^2 + |2v-u|$ to demonstrate "quiet" operation, eliminating idle tones and ensuring energy is minimized when the input is zero (Ward, 2010).

3. Asymmetric Quantization and Information Theory

In communication systems, the use of asymmetric quantizers is critical in maximizing channel capacity, especially in the low-SNR regime. With a symmetric one-bit quantizer, the capacity per unit energy is known to be $1/(\pi\sigma^2)$, incurring a $2/\pi$ penalty over the unquantized Gaussian channel. By shifting the quantizer threshold $$\mathcal{D} = \{y \in \mathbb{R}: y \geq \Upsilon\}$$ and deploying an asymmetric signaling constellation (e.g., pulse position modulation, PPM), the channel achieves the optimal $1/(2\sigma^2)$ capacity per unit energy, recovering the 2 dB loss characteristic of symmetric setups (Koch et al., 2011). The analytical framework involves D-divergence measures under optimal thresholding and demonstrates that capacity recovery is not possible using symmetric quantization, even if the input is discrete (Koch et al., 2012).

4. Asymmetric Schemes in Machine Learning and Neural Network Quantization

Model Parameter and Activation Quantization

The asymmetric mapping of quantization ranges is essential for low-bit quantization in neural networks. Because DNN weights and activations often follow bell-shaped or highly non-uniform distributions, symmetric quantization results in coarse/fine granularity mismatches. Asymmetric schemes deploy a zero-point for translation and a learned or adaptive scaling factor to maximize the utilization of available bit levels:

• Adaptive Step Size Quantization (ASQ) learns an activation-specific scaling $s_a = s \times \beta$, with the per-instance factor $\beta$ predicted by a small adapter module. This mechanism dynamically modulates the interval resolution and is paired with non-uniform weight quantization (such as Power Of Square root of Two, POST) to better fit the empirical weight distribution (Zhou et al., 24 Apr 2025).
• For hardware-friendly non-uniform quantization, quantization points may follow geometric (logarithmic) patterns adapted to data statistics.

Quantization-Aware Training Parameterizations

Parameterizations for learning asymmetric quantization during QAT have strong effects on stability and convergence:

• "Scale/offset" parameterizations are highly sensitive to learning rate and bit width, showing oscillatory or divergent behavior if tuning is imperfect.
• Learning direct min/max bounds ensures that the gradient is automatically scaled by the bit width $k$, stabilizing training.
• Beta/gamma parameterizations ($\beta, \gamma \in \mathbb{R}_+$ with $$s = \frac{\gamma\theta_{\text{max}} - \beta\theta_{\text{min}}}{k}$$) allow proportional and distance-sensitive learning updates, further facilitating rapid and stable QAT (You et al., 25 Apr 2024).

A summary of parameterization methods is presented below:

Parameterization	Formulation	Noted Benefits
Scale/Offset	$s, z$	Direct, but sensitive and unstable
Min/Max	$\theta_{\text{min}}, \theta_{\text{max}}$	Robust to bit-width and LR
Beta/Gamma	$\beta, \gamma$	Fast convergence, distance-aware

5. Asymmetric Quantization in Model Compression and Hardware Acceleration

Layer-wise and Structure-Aware Asymmetry

Compression of transformer KV caches and large-scale DNN weights requires asymmetric assignment of quantization precision:

• In key-value (KV) cache quantization for LLM inference, quantizing key matrices per-channel (to accommodate strong outlier channels) and value matrices per-token (to localize and constrain error) achieves stronger trade-offs between memory, throughput, and quality (Liu et al., 5 Feb 2024).
• AsymKV introduces layer-wise asymmetric bitwidths for keys and values. Up to 75% of decoder layers can be assigned to 1-bit quantization, provided keys in early or sensitive layers are retained at higher precision, with negligible performance loss. Attention output is notably more sensitive to key quantization than to value quantization, due to exponential error amplification by softmax (Tao et al., 17 Oct 2024).

Hardware and Accelerator Designs

Bit-slice matrix multiplication accelerators benefit from asymmetric quantization by increasing the slice-level sparsity and enabling the skipping of both zero and frequent nonzero slices (which become dominant under asymmetric mapping). The Asymmetrically Quantized bit-Slice GEMM (AQS-GEMM) framework employs run-length encoding and zero-point manipulation to maximize HO-slice (high-order) compressibility, dynamically reconfiguring slice bit-widths per layer. The Panacea accelerator implements the necessary dataflows and compression strategies, substantially lowering both compute and memory costs while maintaining accuracy (Kam et al., 13 Dec 2024).

6. Applications Across Signal Processing, Compression, Hashing, and Machine Learning

The design and utilization of asymmetric quantization schemes have broad implications:

• Analog/digital converters: Ensures "quiet" output without idle cycles, reducing audible artifacts and energy consumption (Ward, 2010).
• Source coding and entropy-constrained quantization: Asymmetric two-level quantizers with extended Huffman coding yield improved rate-distortion trade-offs for Laplacian and other asymmetric sources, allowing bit rates near entropy with minimal SQNR loss (Peric et al., 2012).
• Deep hashing and retrieval: Asymmetry in binary hashing (across query and database domains, or using class-structured quantization and dual-label supervision) improves semantic sensitivity and retrieval performance in cross-modal and image search tasks (Wang et al., 2020, Lu et al., 2021).
• Learning vector quantization: Asymmetric prototype averaging, particularly in DTW (Dynamic Time Warping) spaces for time-series, enables stable and interpretable prototype learning, with empirical accuracy and operational gains for fast nearest-neighbor classification in high-dimensional sequence domains (Jain et al., 2017).
• Model compression and large transformer quantization: Asymmetric calibration of each layer by directly matching the quantized output to the original full-precision model output (instead of propagating quantization-induced errors) leads to substantial improvements in language and vision models, even at extremely low bitwidths (Li et al., 3 Apr 2025, Zhang et al., 2023).

7. Impact and Future Directions

Current research demonstrates that exploiting statistical and structural asymmetry in data, neural network architectures, and communication channels can offer both accuracy and computational efficiency not attainable by symmetric quantization alone. Asymmetric schemes are likely to remain a dominant paradigm as demands for aggressive compression, high-fidelity model deployment, and power/throughput optimization increase.

Promising directions include:

• Automated, statistic-driven assignment of asymmetric quantization ranges across layers, channels, and data types.
• Further integration of asymmetric quantization with error-compensated and distillation-based training.
• Deployment-tailored quantization schemes that leverage real-world distribution shifts and data asymmetry.
• Hardware-accelerated frameworks designed natively for asymmetric quantization, encompassing flexible dataflow, run-length compression, and dynamic per-layer reconfiguration.

A comprehensive understanding and application of asymmetry—in both the statistical and operational sense—will drive continued advances in efficient, robust, and high-accuracy digital and neural signal processing.