Temporal Dynamic Quantization

• Temporal dynamic quantization is a method that adjusts quantizer parameters (scale, zero-point, type) over time to match evolving activation distributions in models such as diffusion, recurrent, and spiking networks.
• It employs strategies like per-timestep adaptation and joint time–channel modeling to minimize precision errors such as clipping and rounding, leading to significant FID improvements in low-bit settings.
• By integrating grouping and parallel calibration techniques, this approach enables efficient low-bit deployment with minimal runtime overhead, supporting real-time and edge applications across various modalities.

Temporal Dynamic Quantization

Temporal dynamic quantization is a class of quantization methodologies in which quantization operators—scale factors, zero-points, or even quantizer type—are explicitly parameterized by temporal structure, typically the sampling step index or analogous time-encoding. This paradigm is motivated by the observation that the statistical distribution of activation (and, in some cases, weight) tensors in modern temporal models (diffusion models, recurrent/spiking networks, time-varying control systems) evolves systematically with time or step index, making static quantization schemes (constant interval per layer) grossly suboptimal in both representational fidelity and downstream task performance. Temporal dynamic quantization frameworks span a spectrum from simple per-timestep adaptation of quantization intervals to more elaborate joint time–channel modeling, rotation-based smoothing, and on-the-fly sample-wise quantization, tailored to challenges such as denoising-diffusion, score-based generation, and time-adaptive SNNs.

1. Motivation: Failure Modes of Static Quantization in Temporal Models

In temporally iterated architectures such as diffusion models or time-integrated control systems, the distribution and dynamic range of activations can vary drastically from step to step. In diffusion models, for example, the incremental denoising process leads to nonstationary statistics: low (early) and high (late) timesteps can have dramatically larger or smaller activation ranges compared to the median, and temporal modules (e.g., time-embedding blocks) generate hypersensitive features dependent solely on the current step index (So et al., 2023, &&&1&&&, Huang et al., 2024).

Applying standard static quantization—where a single min–max interval or per-channel scale/zero-point is calibrated offline and fixed for all steps—results in severe precision mismatches:

• At steps where the real dynamic range is much smaller than the static interval, excessive rounding errors arise.
• When the real range exceeds the static interval, catastrophic clipping/truncation errors are induced, mapped directly into loss of semantic trajectory and visual fidelity (e.g., FID degradation by orders of magnitude in low-bit settings) (Liu et al., 2024, Chen et al., 2024).

Temporal dynamic quantization methods were introduced to explicitly recognize and correct this mismatch by embedding time-dependence in the quantization operator, thereby tracking and adapting to the inherent nonstationarity of temporal feature distributions.

2. Core Methodologies in Temporal Dynamic Quantization

While approaches vary across problem domains, recent research has established several principal technical strategies (summarized with primary sources):

A. Per-Timestep Quantizer Parameterization

The canonical form in diffusion models is to replace the usual per-layer quantizer $(s, z)$ with a per-timestep $(s_t, z_t)$ pair, so activation quantization at time $t$ becomes:

$$\bar{x} = \mathrm{clip}\left( \mathrm{round}\left( \frac{x}{s_t} \right) + z_t, \; 0, \; 2^b - 1 \right)$$

$\hat{x} = s_t (\bar{x} - z_t)$

The scale $s_t$ may be determined by explicit function generators of $t$, e.g., small MLPs on Fourier features of $t$ (So et al., 2023), by direct min–max calibration per $t$ (Huang et al., 2023, Huang et al., 2024), or by a hybrid (e.g., piecewise time-grouped intervals as in TGQ (Hwang et al., 6 Feb 2025)).

B. Joint Timestep–Channel Modeling

Temporal statistics interact with channel structure: some channels exhibit wider dynamic ranges or stronger step-to-step drift than others. The timestep-channel adaptive quantization (TCAQ) paradigm introduces a "timestep–channel reparameterization" (TCR), leveraging a channel rebalancing vector $r^s \in \mathbb{R}^C$ such that, after rescaling activations and convolutional kernels, channel activation ranges are equalized across all $t$ and $d$ indices. This minimizes the "worst-case" quantization bottleneck introduced by either timestep or channel outliers (Huang et al., 2024).

C. Non-Uniform or Grouped Temporal Quantization

Rather than allocating a unique quantizer to every timestep, activations are quantized using $G$ groups over $T$ timesteps, with each group assigned its own $(s_g, z_g)$. TGQ (time-grouping quantization) (Hwang et al., 6 Feb 2025) demonstrates that contiguous time groups are able to exploit local step similarity (especially in later, slow-changing denoising regimes), reducing calibration cost without much loss in step-dependent fidelity.

D. Time-Parallel and Blockwise Strategies

Temporal parallel quantization (TPQ), as in DilateQuant (Liu et al., 2024), maintains an array of quantizer parameters across all $T$ time steps. For each mini-batch, the appropriate quantizer is selected according to the step index. Joint training or calibration of all $T$ quantizers (along with weight dilation) is highly parallelizable, resulting in significant reductions in wall-clock quantization and calibration time.

E. Dynamic Quantizer Type Selection

In deep diffusion transformers, late-sampling attention activations develop distinctly heavy-tailed (power-law) distributions, making uniform quantization suboptimal. Dynamically-adaptive quantizers (DAQs) automatically select quantizer distributive form (uniform or logarithmic) on a per-layer, per-timestep basis by modeling activation statistics (e.g., fit power-law tail, log-likelihood-ratio test) (Huang et al., 2024).

F. Temporal Feature Block Quantization

Small, data-independent modules (time-embeddings, temporal MLPs) are quantized by isolating them as "Temporal Information Blocks" (TIBs), and calibrating a separate quantizer (and, if possible, direct minimization of per-feature quantization error) for each $t$. This approach, realized in frameworks like TFMQ-DM (Huang et al., 2023) and Temporal Feature Matters (Huang et al., 2024), is crucial for preserving semantic trajectory in diffusion denoising, and allows for either fast cache-based quantization or fine-grained per-vector optimization.

3. Mathematical Formalism and Implementation

The mathematical definition of a temporal dynamic quantizer is model- and context-dependent, but generic instances include:

• Static per-layer quantizer:

$$Q(x) = s \cdot ( \mathrm{clip}(\mathrm{round}(x / s) + z, 0, 2^b - 1) - z )$$

with $s$ and $z$ fixed across all timesteps.

• Per-timestep dynamic quantizer (So et al., 2023):

$$Q_t(x) = s_t \cdot ( \mathrm{clip}(\mathrm{round}(x / s_t) + z, 0, 2^b - 1) - z )$$

with $s_t = f(\mathrm{enc}(t))$ for some learnable generator $f$.

• Joint timestep–channel reparameterization (Huang et al., 2024):

$$X^{t'}_{:, d} = X^t_{:, d} / r^s_d,\quad W'_{:, d} = W_{:, d} \cdot r^s_d$$

with $r^s_d$ fused via a weighted sum over per-timestep and per-channel maxima.

• TGQ (time-grouped quantization) (Hwang et al., 6 Feb 2025): Group timesteps into $G$ groups, calibrate $s_g$ for each, and assign $Q_g(x_t)$ according to $t \in \mathcal{G}_g$.

Pseudocode and pipeline diagrams for these procedures appear in (So et al., 2023, Huang et al., 2024, Huang et al., 2023, Huang et al., 2024, Hwang et al., 6 Feb 2025, Liu et al., 2024).

4. Empirical Evidence, Trade-offs, and Benchmarks

Comprehensive empirical studies demonstrate the quantitative advantage of temporal dynamic quantization for deployment-quality, low-bit diffusion models. Key outcomes include:

Model	Static PTQ (4 bit)	Temporal Dynamic (4 bit)	Full-Precision	Dataset	Metric (FID ↓, IS ↑)
DDIM	FID > 120	FID ≈ 9 (Liu et al., 2024)	FID ≈ 4	CIFAR-10	FID
TCAQ-DM	FID ≈ 371	FID ≈ 6.4 (Huang et al., 2024)	FID = 4.14	CIFAR-10 (W4A4)	FID
Q-DiT	FID > 250	FID = 15.76 (Chen et al., 2024)	FID = 12.40	ImageNet-256 (W4A8)	FID
TFMQ-DM	FID = 4.42	FID = 3.60 (Huang et al., 2023)	FID = 2.98	LSUN-Bed. (W4A32)	FID
TQ-DiT	FID = 20.53	FID = 8.58 (Hwang et al., 6 Feb 2025)	FID = 4.62	ImageNet-256 (W6A6)	FID

Ablations consistently show that dynamic, fine-grained per-timestep quantization closes the gap to full-precision even under aggressive bit-width reduction, while static quantization at the same precision results in catastrophic mode collapse or severe perceptual degradation.

Temporal adaptive strategies typically incur minimal additional runtime overhead: in many frameworks, dynamic parameters can be precomputed and stored in tables (activation quantization) or their computation costs are negligible compared to main compute (e.g., per-group min/max scans in DiT blocks) (So et al., 2023, Chen et al., 2024). Memory and latency are sometimes further reduced (by up to 3.9× in deployment) due to serial caching of TIB outputs, and calibration costs can be amortized via parallel quantizer updates (e.g., TPQ, TGQ) (Liu et al., 2024, Hwang et al., 6 Feb 2025, Huang et al., 2024).

5. Extensions Across Model Classes and Modalities

Temporal dynamic quantization is not specific to diffusion models, but applies broadly:

• Video diffusion models: Temporal discriminability quantization is crucial, as temporal-feature skewness and inter-channel asymmetry in multi-frame scenarios are exacerbated (Tian et al., 2024).
• Spiking neural networks (SNNs): Temporal-adaptive quantization (e.g., TaWQ) modulates ternary weights dynamically at each step, driven by hidden states mimicking biological astrocytic modulation, enabling highly energy-efficient inference with negligible accuracy drop even for ultra-low-bit hardware (Zhang et al., 14 Nov 2025).
• Control systems: “Dynamic quantizers” parameterized by event-driven or time-varying scale factors (as in (Ren et al., 2020)) drastically reduce the state-space abstraction size and enable scalable approximate bisimulations in nonlinear control.
• Score-based and transformer-based generation: Transformer blocks (DiTs) require specialized grouping (e.g., multi-region quantization co-composed with time grouping), as per (Hwang et al., 6 Feb 2025, Chen et al., 2024).

Additionally, frameworks like TMPQ-DM (Sun et al., 2024) combine non-uniform timestep grouping with layer-wise mixed-precision quantization for joint temporal and structural optimization.

6. Limitations and Practical Considerations

Despite their efficacy, temporal dynamic quantization methodologies have several practical considerations:

• Calibration Complexity: For methods requiring per-timestep or per-(t,x) calibration, the number of quantization parameters can increase substantially with $T$, though approaches such as grouping (TGQ) or sample-wise computation (Q-DiT) mitigate this cost (Hwang et al., 6 Feb 2025, Chen et al., 2024).
• Memory Overhead: Storing per-step or per-group quantization tables or pre-cached temporal features is viable for small $T$ (e.g., $T\lesssim 1000$), but may be prohibitive in extremely long time-horizons, requiring aggressive grouping or hashing.
• Hardware Realization: Some schemes (e.g., per-step rotation matrices (Shao et al., 9 Mar 2025), time-dependent permutations) require custom kernel support or hardware co-design for optimal efficiency.
• Generalizability: For layers with weak temporal correlation, or models where time-dependence is text/condition-dependent, dynamic quantization may offer only modest improvements over static schemes (So et al., 2023).

7. Outlook and Future Directions

Anticipated directions for temporal dynamic quantization research include:

• End-to-End Learned Schedules: Roles for differentiable allocation of bit-widths, quantizer types, and group sizes over both temporal and spatial dimensions (Shao et al., 9 Mar 2025, Liu et al., 2024).
• Integration with Timestep Reduction: Joint optimization over quantization and adaptive generation schedule length (Sun et al., 2024).
• Quasi-Online and Conditionally Parameterized Quantization: Extensions to models where temporal evolution is itself data-dependent (video transformers, adaptive samplers).
• Cross-Domain Generalization: Systematic treatment of temporal quantization in RL, time-series forecasting, sequential control, and hybrid-symbolic abstraction (Ren et al., 2020).
• Hardware–Algorithm Co-Design: Hardware kernels for block-diagonal rotations, on-the-fly quantizer multiplexing, and embedding of large per-step quantization LUTs in on-chip memory.

Taken together, temporal dynamic quantization represents a key enabling technology for bringing temporal deep models—particularly large-scale diffusion generative models and temporal recurrent/spiking architectures—down to the edge or real-time, low-resource deployment, without loss of accuracy or sample fidelity. The field is evolving rapidly, with new strategies for per-timestep, per-channel, and per-sample adaptation emerging in both theory and large-scale empirical evaluations (Huang et al., 2023, Huang et al., 2024, Liu et al., 2024, Chen et al., 2024, Hwang et al., 6 Feb 2025, Zhang et al., 14 Nov 2025, Huang et al., 2024).