Bit Centering & Dynamic Scaling

• Bit Centering and Dynamic Scaling are techniques that recenter and normalize digital values to reduce quantization errors and improve system accuracy.
• Adaptive methods such as per-layer scaling and incremental updates enhance neural network quantization and hardware-aware performance.
• Real-time recalibration of scaling factors in streaming and non-stationary environments ensures robust, efficient digital signal processing.

Bit centering and dynamic scaling encompass a family of techniques that optimize the representation, processing, or adaptation of digital or quantized values by adjusting centering (offset, mean removal, or value anchoring) and scaling (dynamic range, normalization, or precision) operations at the bit level. These techniques play a pivotal role in digital signal processing, numerics, machine learning, quantized neural networks, compressive sensing, data acquisition, feature preprocessing, and hardware-aware optimization. Across these domains, bit centering and dynamic scaling address the challenges of mismatched statistics, changing signal distributions, quantization errors, overfitting, loss of expressivity, and memory or energy constraints.

1. Conceptual Foundations

Bit centering generally refers to recentering numerical representations or model parameters—explicitly or implicitly—to remove mean bias, align with a reference point, or anchor quantization bins. Dynamic scaling involves the adaptive determination or adjustment of scale factors, precision, or ranges based on observed or learned statistics, runtime feedback, or application phase. Both concepts are instantiated as algorithmic primitives or as implicit ingredients embedded in larger learning, inference, or measurement systems. They can take various forms depending on the context:

Domain	Bit Centering Example	Dynamic Scaling Example
Neural network weights	Subtract layer-wise mean before quantization	Per-layer learnable scaling factors
Feature preprocessing	Subtract feature mean (centering)	Incremental variance-based scaling
Data acquisition	Offset ADC counts based on reference/minimum value	Recompute scaling with sensor drift
Signal quantization	Signal mean/offset normalization before binarization	Floating or variable bit widths

Bit centering often interacts with dynamic scaling—e.g., mean subtraction is paired with variance normalization, and block-wise centering is accompanied by block-wise scaling to maximize quantization efficiency.

2. Information-Theoretic and Algorithmic Principles

Theoretical approaches such as those in BICM optimization (Wu et al., 2013) and compressive sensing (&&&1&&&) demonstrate that proper centering and scaling are essential for efficient information transfer and robust recovery under quantization. In mismatched scenarios, the achievable rate is characterized by the Generalized Mutual Information (GMI), which is maximized when the log-likelihood ratios (LLRs) are calibrated using optimal scaling factors. The I-curve for bit channel $i$ is given by

$$I_{qB_i,Y}(s) = 1 - \mathbb{E}_{X,Y}\left\{ \log_2\left[ 1 + \exp(-\mathrm{sgn}(b_i(X)) L_{B_i,Y} s) \right] \right\}$$

and the optimal scaling aligns the I-curve peak to $s=1$ for each channel.

In 1-bit compressive sensing, centering and scaling are essential because quantization removes amplitude information. Universal recovery is possible for bounded-dynamic-range signals if, and only if, the number of measurements scales as $\Omega(R k^{3/2})$ (Bansal et al., 2022), with $R$ the dynamic range, and $k$ the sparsity. Here, centering ensures all nonzero entries are detectable and scaling maintains separability after sign quantization.

In bit-regularized or quantized deep networks, centering (e.g., shifting by $\alpha = \min(W)$) and dynamic scaling (e.g., learnable $\delta$) enable stable, expressive low-precision representations (Raghavan et al., 2017, Daliri et al., 3 Nov 2024), while maintaining a controlled description length as motivated by the MDL principle.

3. Adaptive and Online Methods

Dynamic scaling and bit centering are critical under non-stationary or streaming conditions where global statistical estimation is impossible or suboptimal. In online learning and streaming feature scaling (Bollegala, 2014), centering (incremental mean subtraction) and scaling (incremental standard deviation) are performed using per-sample updates:

$x_k' = \frac{x_k - \mu_k}{\delta_k}$

with

$$\mu_k^{(t)} = \mu_k^{(t-1)} + \frac{x_k^{(t)} - \mu_k^{(t-1)}}{t}$$

$$s_k^{(t)} = s_k^{(t-1)} + (x_k^{(t)} - \mu_k^{(t-1)})(x_k^{(t)} - \mu_k^{(t)})$$

Similarly, in BICM decoding (Wu et al., 2013), an online scaling algorithm incrementally adjusts the LLR scaling factor $s$ per frame by maximizing the empirical GMI via a simple iterative search, eliminating the lag and brittleness associated with static lookup-table-based calibration.

In neural network quantization and training, dynamic precision scaling tracks overflow rates and quantization error percentages to increase or decrease bit widths for weights and activations in real time (Taras et al., 2018). The quantization error metric (e.g., $$E_{(\%)} = |x_{out} - x_{in}| / (x_{in} \times 100)$$) is used to adapt the number of fractional bits according to performance.

4. Hardware and Numerical Optimization

Bit centering and dynamic scaling enable efficiency in embedded systems and hardware accelerators. In measurement systems (Paraschiv et al., 2019), the scaling function is derived analytically by chaining invertible static characteristics from sensor through ADC to digital representation, e.g.,

$q(A_N) = g(f^{-1}(h^{-1}(A_N)))$

with proper centering (e.g., $N_{min}$ adjustment) and scaling for different operating ranges.

In numerical code generation, semantic equation-based frameworks (Adjé et al., 2021) use properties such as the most significant bit (MSB) and number of significant bits (NSB) to guarantee required accuracy with minimal representation cost, solving for optimal per-variable bit allocations via integer linear programming. This allows for bitwise tuning, where centering (through optimized NSB placement) and dynamic scaling (adaptive bitwidths) minimize memory and energy while maintaining numerical fidelity.

Neural network-specific variable-bit encodings such as DyBit (Zhou et al., 2023) employ adaptive allocation of exponent and mantissa lengths per-weight, facilitating variable-value dynamic scaling directly suited to the non-uniform distribution of DNN weights and activations. These representations can be seamlessly exploited by hardware accelerators via dynamic decoding/encoding logic.

5. Scaling Laws and Performance Trade-offs

Systematic studies of quantized model behavior have established scaling laws that relate achievable performance (e.g., accuracy, FID) to the total bit budget, model size, and quantizer properties (Dettmers et al., 2022, Ding et al., 6 Jan 2025, Liu et al., 4 Feb 2025). For LLMs, reducing from 16-bit to 4-bit precision with suitable block sizes and data types provides nearly optimal trade-offs (Dettmers et al., 2022); further reduction (to 3-bit or binary) degrades stability or quality. Block-wise centering and fine-grained scaling (e.g., quantile-based quantization) improve effective bit utilization.

Bit-level scaling laws differ across architectures. Quantized vision generative models exhibit paradigm-specific responses to bit allocation (Ding et al., 6 Jan 2025): language-style models with discrete token spaces tolerate quantization better than diffusion models with continuous representations, due to inherent centering and error absorption in the codebook. Distillation-based methods (e.g., TopKLD) can further elevate scaling behavior by balancing explicit and implicit knowledge transfer.

In extremely low-bit regimes, unified frameworks such as ParetoQ (Liu et al., 4 Feb 2025) reveal that a marked transition exists between 2-bit and 3-bit quantization. For 3 bits and above, fine-tuned models retain proximity to original weight distributions, but at 2 bits and below, weight distributions shift substantially. Straight-Through Estimators and dynamically learned scale factors enable these transitions, and hardware mapping for 2-bit arithmetic delivers practical benefits for bandwidth and speed.

6. Applications and Case Studies

• BICM Receivers: Dynamic LLR scaling, per-frame GMI maximization, and uniform scaling factors yield robust, real-time adaptation to channel variability, improve achievable rates, and enhance frame error rate under realistic mobile scenarios (Wu et al., 2013).
• Online Learning: Incremental feature centering/scaling in logistic regression and passive-aggressive algorithms ensures resilience to concept drift, improving both accuracy and convergence in one-pass datasets (Bollegala, 2014).
• Dynamic Mode Decomposition (DMD): Subtracting the mean or known frequencies prior to linear model fitting isolates physically meaningful modes, improves eigenvalue resolution, and facilitates background/foreground separation in high-dimensional spatiotemporal data (Hirsh et al., 2019).
• Quantized Neural Network Inference: Adaptive, variable-length formats (e.g., DyBit), hardware-aware quantization, and mixed-precision accelerators achieve higher accuracy and up to an 8× speedup at 4 bits, with negligible performance drop (Zhou et al., 2023).
• Model Selection and Cross-Validation: Fast partition-based cross-validation with efficient centering and scaling, avoiding data leakage, enables high-throughput multivariate analysis (PCA, PCR, OLS) with only marginal cost over unprocessed variants (Engstrøm et al., 24 Jan 2024).

7. Limitations, Open Problems, and Future Directions

Although centering and scaling are powerful primitives, their impact is context-dependent. For example, distribution centering is generally ineffective for weight quantization in LLMs (Dettmers et al., 2022), but may be essential for activations or in block-wise quantization regimes. Dynamic exponent formats offer adaptive scaling but are suboptimal at key bit-widths compared to float or quantile schemes (Dettmers et al., 2022). Universal 1-bit recovery is limited by the trade-off between sparsity and dynamic range (Bansal et al., 2022).

Open research avenues include:

• Extending real-time adaptive scaling to non-Gaussian, heavy-tailed, or multimodal distributions in streaming data.
• Developing formal scaling laws for low-bit quantization in architectures beyond transformers and diffusion models, with an emphasis on error propagation and feedback-driven scale adaptation.
• Hardware-software co-design of quantization, centering, and dynamic scaling primitives for next-generation edge devices.
• Advanced distillation and representational re-centering strategies to mitigate accuracy degradation under severe bit constraints, particularly in complex or multi-modal generative tasks (Ding et al., 6 Jan 2025).

Bit centering and dynamic scaling thus constitute foundational building blocks for reliable, efficient, and robust digital systems spanning communication, sensing, learning, and inference. Their continued refinement and integration are crucial for deploying adaptive, scalable, and resource-constrained intelligent systems.