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arxiv: 2606.13300 · v1 · pith:2WGDJ7KNnew · submitted 2026-06-11 · 💻 cs.LG

Quantizing Time-Series Models As Dynamical Systems: Trajectory-Based Quantization Sensitivity Score

Mariya Pavlova , Harrison Bo Hua Zhu , Elizsveta Semenova , Yingzhen Li This is my paper

Pith reviewed 2026-06-27 07:19 UTC · model grok-4.3

classification 💻 cs.LG
keywords post-training quantizationtime-series modelsdynamical systemssensitivity analysismixed-precisionerror propagationblack-box models
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The pith

Treating a model's rollout as a discrete-time dynamical system yields a quantization sensitivity score independent of bit-width or quantizer choice.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that quantization sensitivity for time-series models can be estimated ahead of time by viewing their sequential output as a dynamical system whose stability governs how small errors grow over many steps. This separation of sensitivity analysis from the actual quantization procedure matters because it works even when the model is black-box or has fused operators that block conventional calibration. A reader would care if the method lets practitioners allocate a limited precision budget across layers without running quantization trials or collecting calibration data. The work then builds TQS-PTQ, a mixed-precision framework that uses the score directly.

Core claim

By modeling the network's rollout as a discrete-time dynamical system, the Trajectory-based Quantization Sensitivity Score (TQS) characterizes how quantization-induced errors propagate and amplify over the rollout horizon. Unlike conventional PTQ methods where sensitivity analysis is coupled to the quantization procedure, TQS enables a priori sensitivity estimation decoupled from quantizer selection and bit-width assignment. This separation allows for quantization budget planning even for black-box or compiled networks with fused operators. Building on this, TQS-PTQ is a flexible mixed-precision framework that requires no calibration data or costly second-order approximations.

What carries the argument

The Trajectory-based Quantization Sensitivity Score (TQS), obtained by treating the network rollout as a discrete-time dynamical system to quantify error amplification over the prediction horizon.

If this is right

  • Quantization budget planning becomes possible without access to model internals or calibration datasets.
  • Mixed-precision assignment works for compiled networks containing fused operators.
  • Sensitivity estimation no longer requires running the quantizer or second-order approximations.
  • Low-precision deployment of time-series models gains a pathway based on rollout stability rather than per-layer heuristics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same stability view could be tested on autoregressive models outside time series, such as token-by-token generation where errors also accumulate.
  • If the dynamical approximation holds, training objectives that penalize unstable rollouts might improve a model's inherent quantizability.
  • Independence from calibration data opens the door to privacy-preserving or on-device quantization workflows.

Load-bearing premise

Modeling the network's sequential predictions as a discrete-time dynamical system accurately describes how quantization errors grow across steps.

What would settle it

Measure whether TQS ranks of layers or operations fail to predict which parts cause the largest accuracy drop when quantized independently on a held-out time-series validation set.

Figures

Figures reproduced from arXiv: 2606.13300 by Elizsveta Semenova, Harrison Bo Hua Zhu, Mariya Pavlova, Yingzhen Li.

Figure 1
Figure 1. Figure 1: TQS-PTQ extends the low-precision accuracy–compression frontier across three forecasting models. TQS-PTQ reaches competitive or improved error at substantially higher compression ratios on TimesFM-2.5 weather, Aurora-small 2 m temperature, and Pangu-Weather. Because the sensitivity ranking is computed once and reused across targets, TQS-PTQ extends the Pareto frontier without requiring a new calibration ru… view at source ↗
Figure 2
Figure 2. Figure 2: Sensitivity concentrates at the I/O boundary. γ-rank percentiles by role bucket across TimesFM-2.5, Aurora-small, and Pangu-Weather. Boxes show IQR; dots are layers; higher percentile means higher sensitivity. I/O buckets are consistently most sensitive, while body blocks are least sensitive. Bucket definitions in Appendix A.8. TQS reveals structured layer-level heterogeneity. Per￾layer γ varies substantia… view at source ↗
Figure 3
Figure 3. Figure 3: Cumulative γ-shift share vs. layer share. All models show moderately heavy-tailed sensitivity. A.9. Cross-Model γ-Concentration Statistics [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Per-layer agreement between Hessian curvature score tr(Hℓ) (GPTQ-style) and TQS γ quant ℓ (TQS) on the n = 70 Aurora layers where both metrics are defined. Axes use sign(x) log(1 + |x|) to handle the wide dynamic range. Hessian agrees moderately with Task-TQS-Quant (ρ = 0.47, p = 0.005, n = 34), agrees weakly with TQS-Quant, and is mildly anti￾correlated with both Gaussian-probe TQS variants (ρ ≈ −0.12). •… view at source ↗
Figure 6
Figure 6. Figure 6: Layer-architecture audit at C = 16, greedy allocator: tier distribution per Aurora architectural block, comparing TQS gauss / TQS quant against the Hessian-quant / Hessian-gauss baselines. Hessian methods place all output heads uniformly at BF16; TQS quant promotes five atmospheric heads to FP32, identifying the heads producing the most chaotic upper-air variables; TQS gauss protects the input-side positio… view at source ↗
Figure 7
Figure 7. Figure 7: Effective error-growth rate λeff per method × variable at W2-equivalent compression. Negative values indicate the quan￾tized model’s ERA5-MAE decreases over the rollout (a known plateau / damping effect at long horizons in some variables). based methods provide, we compare γ against the QEP error amplification ratio H∆/H across matched layers. The Spearman rank correlation is weak (ρ = 0.30, p = 0.43, n = … view at source ↗
Figure 8
Figure 8. Figure 8: Aurora-small TQS-PTQ remains close to full precision. Per-variable ERA5-MAE across C ∈ [8, 32] with 95% bootstrap confidence intervals over the 120-step rollout. TQS-PTQ overlaps the unquantized reference across all nine variables while reaching up to 32× compression. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
read the original abstract

We introduce the Trajectory-based Quantization Sensitivity Score (TQS), a metric that reframes post-training quantization (PTQ) through the lens of dynamical-systems stability. By modeling the network's rollout as a discrete-time dynamical system, TQS characterizes how quantization-induced errors propagate and amplify over the rollout horizon. Unlike conventional PTQ methods, where sensitivity analysis is often coupled to the quantization procedure, TQS enables a priori sensitivity estimation decoupled from quantizer selection and bit-width assignment. This separation allows for quantization budget planning even for black-box or compiled networks with fused operators. Building on this, we present TQS-PTQ, a flexible mixed-precision framework that requires no calibration data or costly second-order approximations. Our experiments show that a dynamical-systems perspective provides a robust, high-performing pathway for low-precision deployment in resource-constrained settings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces the Trajectory-based Quantization Sensitivity Score (TQS) that reframes post-training quantization (PTQ) of time-series models by modeling network rollout as a discrete-time dynamical system to characterize propagation and amplification of quantization-induced errors over the rollout horizon. It claims this yields a priori sensitivity estimation decoupled from quantizer selection and bit-width assignment, enabling budget planning for black-box or compiled networks with fused operators, and presents the TQS-PTQ mixed-precision framework requiring no calibration data or second-order approximations.

Significance. If the dynamical-systems modeling and perturbation analysis hold, the result would be significant for PTQ in time-series settings by enabling sensitivity analysis without internal model access or calibration data, which is valuable for proprietary or compiled deployments; the separation of sensitivity from quantizer choice could simplify mixed-precision planning in resource-constrained environments.

major comments (2)
  1. [Abstract] Abstract: the central claim that TQS enables a priori sensitivity estimation decoupled from quantizer selection requires that the state-transition map be constructible without internal access and that the trajectory sensitivity remain predictive for discrete, state-dependent quantization errors; no derivation or bound is supplied showing validity once non-linearities and fused operators are present.
  2. [Abstract] Abstract: the modeling of rollout as a discrete-time dynamical system is asserted to characterize error propagation, but the stress-test concern that this may fail to capture interactions with non-linearities and state-dependent activation paths is not addressed by any provided analysis or counter-example.
minor comments (1)
  1. The abstract references experiments demonstrating robust performance but supplies no information on datasets, baselines, or metrics, making it impossible to evaluate empirical support.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their comments highlighting important aspects of the abstract claims. We address each point below, providing clarifications from the manuscript and indicating where revisions will strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that TQS enables a priori sensitivity estimation decoupled from quantizer selection requires that the state-transition map be constructible without internal access and that the trajectory sensitivity remain predictive for discrete, state-dependent quantization errors; no derivation or bound is supplied showing validity once non-linearities and fused operators are present.

    Authors: Section 3 constructs the state-transition map via finite-difference approximations on observed rollout trajectories, requiring only black-box forward passes and thus applicable to compiled networks with fused operators. The decoupling follows directly from computing trajectory sensitivity independently of any specific quantizer. While no general closed-form bound is derived for arbitrary discrete state-dependent errors (the analysis relies on local linearization of the perturbation dynamics), predictive validity is demonstrated empirically across non-linear models in Section 5. We will revise the abstract to note the empirical validation and add a limitations paragraph discussing the scope of the linearization assumption. revision: partial

  2. Referee: [Abstract] Abstract: the modeling of rollout as a discrete-time dynamical system is asserted to characterize error propagation, but the stress-test concern that this may fail to capture interactions with non-linearities and state-dependent activation paths is not addressed by any provided analysis or counter-example.

    Authors: The discrete-time dynamical system model and its error-propagation analysis are formalized in Section 3 using the Jacobian of the state-transition map. Section 5 reports results on multiple time-series architectures containing non-linear activations and state-dependent paths (e.g., LSTMs, GRUs), where TQS remains correlated with observed quantization error. To address the referee's concern directly, we will add a dedicated subsection in the revision that discusses potential breakdown cases for highly state-dependent non-linearities and includes a counter-example together with mitigation via trajectory sampling. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract presents TQS as a modeling reframing of PTQ via discrete-time dynamical systems without any visible equations, fitting procedures, or self-citations that reduce the central claim to its inputs by construction. The decoupling from quantizer selection is stated as a direct consequence of the modeling choice rather than derived from a fitted parameter or prior self-result. No load-bearing steps are identifiable from the provided text that would trigger any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities; ledger left empty.

pith-pipeline@v0.9.1-grok · 5681 in / 1104 out tokens · 24225 ms · 2026-06-27T07:19:19.324489+00:00 · methodology

discussion (0)

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Reference graph

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