Distribution-Aware Quantization

• Distribution-aware quantization is a technique that customizes quantizer parameters to the statistical properties of data such as weights and activations in neural networks.
• It employs models like the generalized gamma distribution, empirical quantile codebooks, and transformation techniques to optimize quantization for diverse and non-uniform data distributions.
• This approach has demonstrated significant performance gains, achieving improvements in MSE and PSNR in ultra-low-precision, federated learning, and vision transformer applications.

Distribution-aware quantization refers to a family of quantization methods in which the quantizer parameters, codebooks, or transformation strategies are explicitly designed or adapted to account for the empirical or assumed probability distribution of the data being quantized—weights, activations, gradients, or model updates in neural networks. Unlike traditional uniform quantization, which assumes a flat distribution and applies an identical grid across all data, distribution-aware quantization leverages statistical properties or fitted models of the underlying data to minimize quantization error, especially in demanding regimes such as ultra-low-precision neural networks, imbalanced federated learning, or specialized hardware post-training quantization.

1. Statistical Motivation and Definitions

Distribution-aware quantization aims to address the mismatch between the intrinsic data distributions present in neural representations (weights, activations, gradients) and the parametric assumptions of standard quantizers. In convolutional and transformer models, empirical statistics reveal wide heterogeneity:

• Feature maps and activations often display per-channel variation, heavy-tailed, skewed, or multimodal distributions (e.g., Gaussian, exponential, Laplacian, or even power-law for post-Softmax outputs in ViTs) (Hong et al., 2023, Yang et al., 2024).
• Weights in INRs and deep MLPs can alternate between uniform, unimodal, or bimodal across layers (Zhou et al., 19 Aug 2025).
• Local model updates in federated learning exhibit approximately normal distributions after standardization, but client heterogeneity induces local variance (Kim et al., 30 Jun 2025).

The rationale is that minimization of quantization mean squared error (MSE), rate-distortion, or signal-to-quantization-noise ratio (SQNR) under a mismatched quantizer is always suboptimal; the Bayes-optimal quantizer utilizes knowledge of $p(x)$, as shown in both theoretical and empirical settings (Jia et al., 22 Oct 2025, Kim et al., 2018).

2. Distribution Fitting and Parameter Estimation Methods

Key methodologies for building distribution-aware quantizers involve first fitting a statistical model to the data to guide codebook and parameter selection.

• Generalized Gamma Distribution (GGD) Fitting: CNN activations, especially after ReLU, are well modeled by the GGD:

$$p(x) = \frac{\beta}{2\alpha\Gamma(1/\beta)} \exp\left(-|x/\alpha|^\beta\right).$$

This parametric fit allows calculation of asymptotically optimal quantization step sizes for a specified number of bits via closed forms in terms of GGD parameters (Kim et al., 2018).

• Empirical Quantile Codebooks: Weight distributions are partitioned into bins by sorting and assigning codebook entries to quantiles, so that each quantization interval carries approximately equal probability mass (Jia et al., 22 Oct 2025).
• Adaptive or Learned Range Estimation: Post-hoc sampling of activations (with outlier trimming) or per-channel percentile estimation (e.g., $\gamma$-percentile of activations for determining bounds) focuses the quantizer range on the distribution's bulk, disregarding tails that would otherwise inflate the dynamic range and erase effective resolution (Chen et al., 5 Oct 2025).
• Transformation-based Unification: Orthogonal transforms (e.g., Hadamard) are applied to convert diverse or non-Gaussian distributions into bell-shaped distributions, after which a uniform quantizer suffices (Zhou et al., 19 Aug 2025).

3. Quantizer Construction and Mapping Strategies

A diversity of mapping, codebook formation, and truncation schemes are used to match the quantizer to the distribution:

• Non-Uniform Quantization: Codebooks explicitly optimize MSE under the fitted prior, e.g., solving for quantization levels $\{q_r\}$ and bins $\{[u_r, u_{r+1})\}$ that minimize

$\sum_r \int_{u_r}^{u_{r+1}} (x - q_r)^2 p(x) dx$

for $p(x)$ given by the data model (Kim et al., 30 Jun 2025, Jia et al., 22 Oct 2025).

• Exponential Codebooks (POT/POST): For bell-shaped or log-dense weight distributions, weights are quantized to levels of the form $\pm\alpha (2)^{k}$ (POT) or $\pm\alpha (\sqrt{2})^{k}$ (POST), the latter softening near-zero clustering and better matching empirical distributions (Zhou et al., 24 Apr 2025).
• Outlier Preservation and Mixed-Precision: Distribution-aware post-training quantization schemes often separate out the extreme subset (e.g., top/bottom $0.5\%$) of weights, storing them in higher precision (FP16), while quantizing the central mass more aggressively (Chen et al., 5 Oct 2025).
• Transformation before Quantization: When distributions are highly non-uniform, a nonlinear or orthogonal transformation (e.g., modulo folding for signals (Chemmala et al., 2024); Hadamard for neural data (Zhou et al., 19 Aug 2025)) is applied, homogenizing the marginal distributions and making non-uniform quantization unnecessary or more hardware feasible.
• Dynamic Scaling and Online Codebook Adaptation: Exponential moving average (EMA) and quantile-tracking allow codebooks for weights or scaling factors for activations to track distributional drift during quantization-aware training (Jia et al., 22 Oct 2025, Zhou et al., 24 Apr 2025).

4. Specialization for Tasks, Architectures, and Representations

Distribution-aware quantization is adapted to specific model architectures and use-cases:

• Super-Resolution and Diffusion Models: Outlier-aware quantizers, groupwise splitting via k-means, and prompt-adaptive log quantization for attention matrices are used to prevent quality collapse in sub-8-bit quantization of U-Nets and attention-based architectures (Ryu et al., 8 Jan 2025, Chen et al., 5 Oct 2025, Hong et al., 2023).
• Federated Learning: Distribution-aware non-uniform quantization of model updates is combined with weight standardization, targeting local Gaussianity to reduce uplink communication and maintain global model accuracy under ultra-low-bit constraints (Kim et al., 30 Jun 2025).
• Vision Transformers: Softmax activations exhibit power-law characteristics; specialized quantizers (e.g., Tan Quantizer, which warps the quantization space to allocate higher resolution to both tails) and MAD-guided scaling mitigate the loss due to naive uniform quantization (Yang et al., 2024).
• Hardware Efficiency: Distribution-aware transformations such as the Hadamard transform simultaneously standardize distributions across all layers, enabling a single uniform quantizer and simplifying FPGA or ASIC pipelines (Zhou et al., 19 Aug 2025). Modulo folding enables nearly-universal, distribution-agnostic quantization for signal processing applications (Chemmala et al., 2024).

5. Analysis, Error Metrics, and Theoretical Guarantees

• Rate–Distortion Perspective: Distribution-preserving quantization (DPQ) constrains reconstruction marginals to the source law, yielding a distribution-preserving rate–distortion function (DP-RDF) that quantifies the fundamental performance limits of DPQ. In the high-rate limit, these bounds converge to classical rate–distortion; at low rates, they preserve perceptual naturalness by enforcing statistical similarity (Li et al., 2011).
• Error Metrics: Most schemes report mean-squared error, PSNR improvements over baselines, or quantization loss, always demonstrating that distribution-aware quantization effects superior SQNR at comparable or lower bit-width (Kim et al., 2018, Kim et al., 30 Jun 2025, Chen et al., 5 Oct 2025).
• Ablations and Quantitative Comparison: Empirical studies show that outlier-aware and distribution-matching schemes recover up to $4$–$6$ dB PSNR over generic PTQ baselines at $4$-bit quantization, close or even surpass full-precision accuracy in ImageNet classification at $4$-bits, and preserve alignment metrics in text-to-image models otherwise destroyed by naive quantization (Chen et al., 5 Oct 2025, Zhou et al., 24 Apr 2025, Ryu et al., 8 Jan 2025, Zhou et al., 19 Aug 2025).

6. Implementation and Practical Guidelines

A synthesis of best practices for implementing distribution-aware quantization schemes emerges:

Step	Key Techniques	Purpose / Effect
Distribution Fitting	GGD, empirical quantiles	Identify shape, tails, bulk
Codebook/Bound Initialization	Quantiles, per-channel sample	Equal-mass bins, outlier insulation
Transformation	Hadamard, modulo, nonlinear	Distribution unification or “flattening”
Outlier Handling	FP16 storage, group-wise	Avoid resolution loss due to tails
Online/Batch Adaptation	EMA, calibration, adapters	Track distributional shift, optimize for current data
Calibration/Fine-tuning	Frequency-aware losses, band-weighted objectives	Target perceptual or task-centric reconstruction
Hardware Considerations	LUTs, bitshifts, resource folding	Efficient post-quant or mixed-precision inference

In practical deployments, calibration data of only modest size (e.g., $\sim$200–1000 samples) suffices, and additional storage or compute for outliers or codebooks is negligible compared to overall memory and bandwidth savings. Sensitivity analysis or bit allocation heuristics enable effective mixed-precision assignments and layerwise codebook optimization (Jia et al., 22 Oct 2025, Kim et al., 30 Jun 2025).

7. Limitations, Open Problems, and Theoretical Extensions

Despite documented empirical and hardware efficiency gains, several outstanding challenges and research directions remain:

• Distribution Estimation: Most methods assume static or quasi-stationary distributions, but online adaptation for non-i.i.d., time-varying, or federated settings remains challenging (Kim et al., 30 Jun 2025, Jia et al., 22 Oct 2025).
• Optimality Theory: Theoretical bounds such as the DP-RDF validate information-theoretic limits but practical, low-complexity DPQ constructions for arbitrary sources or memory processes are far from fully developed (Li et al., 2011).
• Perceptual Distortion Measures: While energy-based (MSE, SQNR) objectives dominate, integration of perceptual metrics or learned losses into distribution-aware frameworks is not yet standardized (Chen et al., 5 Oct 2025).
• Universal/Blind Quantization: Modulo folding or compander-based approaches work “blind” to the source, but at the expense of oversampling or increased decoder complexity (Chemmala et al., 2024).
• Algorithmic Complexity: Fine-grained group-wise or per-channel adaptation increases calibration or inference cost, though hardware-specific optimizations (e.g., Hadamard folding into MACs) mitigate this for some architectures (Zhou et al., 19 Aug 2025).

In conclusion, distribution-aware quantization provides foundational strategies, both theoretically and in large-scale practice, for closing the gap between low-precision efficiency and statistical (or perceptual) fidelity in modern neural and signal processing models. Across domains, it unlocks new regimes of mixed-precision, low-power, or edge deployment previously inaccessible to classical uniform quantization approaches.