CoSTI: Consistency Models for (a faster) Spatio-Temporal Imputation

Bel\'en Vega-M\'arquez; Isabel A. Nepomuceno-Chamorro; Javier Sol\'is-Garc\'ia; Juan A. Nepomuceno

REVIEW 3 major objections 4 minor 58 references

CoSTI: Consistency Models for (a faster) Spatio-Temporal Imputation

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read CoSTI adapts consistency models to multivariate time series imputation, matching diffusion-based accuracy while cutting inference time by up to 98%.

desk verdict First consistency-model adaptation for MTSI, but the headline 98% speedup is compromised by an unresolved contradiction between Algorithm 2's 100-pass median and the text's 'single forward pass' claim. read the letter →

arxiv 2501.19364 v2 pith:PTB24QYP submitted 2025-01-31 cs.LG cs.AI

Javier Solís-García , Belén Vega-Márquez , Juan A. Nepomuceno , Isabel A. Nepomuceno-Chamorro This is my paper · ORCID

classification cs.LGcs.AI

keywords multivariatetimeseriesimputationconsistencymodelstraininggenerativespatio-temporaldatadiffusionmissingreal-timeinference

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

CoSTI adapts consistency models — generative models trained to map a noised input directly back to the clean signal in one step — to multivariate time series imputation. The paper argues that with conditional inputs (a linear-interpolation placeholder, the missing-value mask, and a fixed spatial graph) and a curriculum over noise levels, a consistency model can match the imputation accuracy of diffusion-based state-of-the-art while reducing inference time by up to 98%. If this is right, generative imputation becomes practical for real-time ICU monitoring, traffic control, and energy systems, where the iterative denoising of DDPMs has been too slow. The evidence spans six datasets with point, block, and natural missingness, and includes a downstream node-prediction check.

What carries the argument

The central object is the consistency function $\boldsymbol{f}_\theta(\mathbf{x}_\sigma, \sigma)$, parameterized as $c_{\text{skip}}(\sigma)\mathbf{x} + c_{\text{out}}(\sigma) F_\theta(\ldots)$ and implemented by a U-Net with spatio-temporal feature extraction modules (STFEM), noise-extraction modules (NEM), and a separate conditioning branch that ingests the linear-interpolation estimate, the mask, and the graph. The training objective enforces self-consistency between adjacent noise levels of the PF-ODE using a Pseudo-Huber metric, while a linear curriculum scheduler raises the number of noise levels from 10 to 200 during training. This machinery replaces the diffusion reverse chain: one consistency evaluation maps noise to data where a DDPM would need $T = 50$ or $T = 100$ iterative denoising steps.

What would settle it

Run Algorithm 2 end-to-end on METR-LA's test set, timing all 100 forward passes plus the median aggregation on the same GPU, and compare with one complete 50-step diffusion reverse chain; if the per-window time ratio is not close to 1:50, the up-to-98% reduction claim is refuted.

Watch

Extended reading notes

Core claim

The core discovery is that consistency training transfers from image generation to structured spatio-temporal imputation without needing a pretrained diffusion teacher. CoSTI learns a consistency function $f_\theta$ that takes a $\sigma$-noised observation together with conditional information and outputs the clean imputation, trained so that outputs for adjacent noise levels along the probability-flow ODE agree. At inference, each sample is produced in one (or two) network evaluations, and the final deterministic imputation aggregates 100 such samples by element-wise median. The paper reports per-dataset inference times of 0.005 to 0.48 hours versus 0.22 to 18.19 hours for DDPM-based models, with MAE and MSE close to those baselines across all six datasets.

Load-bearing premise

The speed claim rests on the reported inference time covering the full imputation procedure in Algorithm 2, which runs 100 stochastic forward passes per window and takes the median; if Table 3 counts only one of those passes, the 98% reduction would shrink by roughly a factor of 100 and the central comparison would not hold.

Editorial extensions

If this is right

Generative imputation can operate at speeds suitable for real-time ICU monitoring and traffic control rather than offline batch processing.
Because CoSTI is trained from scratch with consistency training, it avoids the extra cost of first training a diffusion model and then distilling it.
The two-step variant CoSTI-2 offers a direct accuracy/speed dial, improving MAE and MSE on most datasets with only one additional forward pass.
On the reported accuracy-time trade-off, CoSTI occupies the Pareto frontier, so no evaluated method is simultaneously faster and more accurate in the tested scenarios.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same conditional consistency formulation should transfer to forecasting by treating the future window as the missing block and using observed history as conditioning; this reuses Algorithm 1 without architectural changes.
The 100-sample median used for deterministic imputation could double as an uncertainty estimator: the spread of the samples can be summarized as confidence intervals for clinical or traffic decisions, which the paper does not report.
The fixed-graph assumption is the least-transferable design choice; on networks with changing topology, a learnable or time-varying adjacency matrix would be the natural next test, and the ablation results indicate the conditioning head would need to absorb that change.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is the first application of consistency models to multivariate time series imputation, and the engineering is mostly solid. But the central speed claim is built on a timing ambiguity that the paper does not resolve. If the reported times count only one of Algorithm 2's 100 forward passes, the 98% reduction is roughly 100x overstated.

What's genuinely new: CoSTI brings consistency training to MTSI, with a conditional U-Net architecture borrowing from CSDI/PriSTI/TIMBA. The evaluation is broad: six datasets, multiple missingness scenarios, sensitivity analysis, downstream task, ablations of the conditional head, STFEM, NEM, and curriculum schedule. The ablations are informative and the best-config choices (linear scheduler, AdamW-ScheduleFree) are backed by experiments. Code and Docker are promised. That's real work.

The problem is the timing. Algorithm 2 runs 100 stochastic imputations per window and takes a median. Section 5.4.1 says Table 3 was obtained "using a single sampling step," and Section 4.3 claims "a single forward pass." Those cannot both describe the measured pipeline. If Table 3 times exclude the other 99 passes, the per-window cost is 100x higher, and the step-count advantage over T=50/100 diffusion steps essentially disappears. For AQI-36, CSDI uses T=100 and CoSTI with 1 step still needs 100 samples, so equal forward passes; the reported 44x speedup would then require a 44x per-pass advantage, which is not argued. The authors also don't say whether baseline diffusion timings include one sample or many, so the comparison might be asymmetric. This is not a minor cosmetic inconsistency; it's the load-bearing pillar of the paper.

Accuracy-wise, the paper claims "comparable," but CoSTI is noticeably worse on PEMS08 (MAE 11.09 vs 9.87 for CSDI) and ETTh1 (0.38 vs 0.33), and closer elsewhere. The 2-step variant helps but doesn't close the gap. Not disqualifying, but the "on par" language is too strong.

Other notes: sigma_i2 is tuned per dataset on the same benchmarks, a mild overfitting to the test sets, but common in this literature. Self-citations to TIMBA are appropriate; no citation problem.

Who is this for? Anyone working on fast generative imputation or consistency models for time series. It deserves peer review; the idea is worth pursuing. But as it stands, I would not accept the speed claim without a clear statement of the number of forward passes in the timed pipeline and a matching code check. If the timing is confirmed to include all 100 passes, the contribution stands; if not, the headline fails. My recommendation: send to referees, require the timing clarification and a rerun if necessary.

Referee Report

3 major / 4 minor

Summary. The paper proposes CoSTI, an adaptation of consistency models to multivariate time series imputation. The method trains a consistency model with a spatio-temporal U-Net architecture, conditional information, curriculum learning over the number of noise levels, and a deterministic imputation stage based on the median of 100 stochastic forward passes. The authors report imputation accuracy comparable to DDPM-based baselines (CSDI, PriSTI, TIMBA) across several datasets and missing-data scenarios, and claim a reduction of inference time of up to 98% relative to the diffusion baselines. The paper also includes benchmark comparisons, sensitivity analyses, ablations, and a downstream task evaluation.

Significance. If the main claim holds, CoSTI would be the first application of consistency models to multivariate time series imputation and would offer a practically useful accuracy/speed trade-off in a domain where diffusion models are accurate but slow. The paper has clear strengths: the code and Docker setup are released, the experimental coverage is broad (six datasets, multiple missingness scenarios, ablations, downstream evaluation), and the accuracy numbers are mostly competitive with the diffusion baselines. However, the central speed claim rests on a timing comparison whose protocol is internally inconsistent with the published sampling algorithm, and this issue must be resolved before the contribution can be assessed.

major comments (3)

[§4.1.4, §4.3 (Algorithm 2), §5.4.1 (Table 3)] There is a direct internal contradiction between the timed inference protocol and the sampling algorithm. Algorithm 2 runs j=1..100 stochastic forward passes per window and then takes an element-wise median, and Section 4.1.4 states N=100 for this deterministic imputation. Section 4.3, however, claims that 'CoSTI requires only a single forward pass' and derives a ~T-fold speedup over T-step diffusion models. Section 5.4.1 states that Table 3 was obtained 'using a single sampling step'. If the reported CoSTI times include only one of the 100 passes, then the true per-window work is about 100 model evaluations; the step-count advantage over T=50 or T=100 diffusion steps disappears (and for AQI-36, where CSDI uses T=100, the forward-pass counts are equal). In that case the reported 44x speedup on AQI-36 would have to come from a per-forward-pass architectural advantage that is neither claimed nor demonstrated. The authors must state unambiguously whether Table 3 reports end-to-end time for the full Algorithm 2 procedure or for a single forward pass, and must report the corresponding baselines under the same protocol.
[§5.4.1 (Table 3), §4.3] The complexity analysis in Section 4.3 is not a valid substitute for an empirical timing comparison once Algorithm 2 is used. Even if a single consistency-model forward pass is cheaper than one diffusion step, the paper must report the total number of model evaluations per imputed window for each method, including the 100 samples used for the median and the number of stochastic samples used for each diffusion baseline. At present, the text does not state whether the CSDI/PriSTI/TIMBA timings include one stochastic imputation sample or several, so the comparison may be asymmetric in either direction. The step-count argument in Section 4.3 and the '1/T times faster' statement in Section 5.4.1 are only valid if every method is measured with the same number of stochastic samples per window.
[§5.3 (Table 2), §5.4.1 (Tables 4-5)] The dataset-specific second noise level sigma_i2 is described as 'experimentally determined' and its values are reported in Table 2, but the selection procedure is not specified. If sigma_i2 was tuned using the same test scenarios that are later reported in Tables 4 and 5, the accuracy advantage of CoSTI-2 (and the claimed 'controllable trade-off') could be optimistically biased. The authors should state whether sigma_i2 was chosen on validation data only, and if not, how the reported CoSTI-2 numbers should be interpreted.

minor comments (4)

[Eq. (10)] In Eq. (10), the second displayed formula is written as c_in(σ) = ln(σ)/4, but from the context it should be c_noise(σ) = ln(σ)/4; this appears to be a typo.
[Eq. (4)] The expression for c_skip(σ) appears to be missing a square in the denominator: the standard form is σ_data^2 / ((σ - σ_min)^2 + σ_data^2), while the paper writes (σ - σ_min) without the square. Please check and correct.
[Algorithm 2] The comment 'keep only missing values' next to the line ̂X_t ← [ ̃X_t ⊙ M_t + ̂X_t ⊙ (1 - M_t)] is misleading: the operation keeps the observed values from the input and the imputed values at missing positions, which is the opposite of keeping only missing values.
[§5.4.1] The statement that CoSTI is 'approximately 1/T times faster' than diffusion models conflates step count with wall-clock time; it should be phrased as a reduction in the number of network evaluations, with wall-clock speedup depending on per-forward-pass cost and any overhead from the 100-sample median loop.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: the accuracy and speed claims are empirical measurements, and the consistency-training objective follows external consistency-model theory; the 100-sample median timing ambiguity is a measurement concern, not a circularity.

full rationale

CoSTI makes two central claims: comparable imputation accuracy to DDPMs and up to a 98% reduction in inference time. Neither claim is derived from a quantity defined to equal it. The accuracy results in Tables 4-6 are direct empirical evaluations on fixed benchmarks; they do not follow from the consistency-training loss in Eq. (6) or the parameterization in Eq. (3) by construction. The speed claim is an empirical runtime comparison (Table 3) plus an asymptotic step-count argument in Section 4.3. That argument counts one consistency-model forward pass versus T diffusion steps, which is a standard complexity comparison rather than a definitional identity. There is, however, an internal ambiguity worth flagging explicitly: Section 5.4.1 says Table 3 was obtained "using a single sampling step" and Section 4.3 says "CoSTI requires only a single forward pass", while Algorithm 2 runs "for j = 1 to 100" stochastic forward passes and then forms "X_t <- Median(S)". If the reported CoSTI times include all 100 passes, the per-window work is much larger than "single forward pass"; if they include only one pass, the times do not correspond to the deterministic imputation procedure described by Algorithm 2. This is a measurement/protocol inconsistency that affects the magnitude and fairness of the 98% speedup, but it is not a reduction of a prediction to its inputs by construction, so it does not meet the criteria for circularity. The paper's self-citations are not load-bearing circularity: Section 5.2 sources CSDI, PriSTI, and TIMBA numbers from the authors' previous work [25], but those are external benchmark measurements that are reproducible and are not used to define CoSTI's predictions. Per-dataset hyperparameters such as sigma_i2 are experimentally selected, but this is ordinary tuning rather than a fitted parameter renamed as a prediction. No uniqueness theorem is imported from the authors, no ansatz is smuggled in through self-citation, and no known result is merely renamed. Overall, the derivation chain is self-contained in the sense relevant to circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The method relies on standard consistency model theory and domain assumptions about fixed graph structure and synthetic missingness transfer. Free parameters include the per-dataset second noise level and curriculum endpoints tuned on the benchmarks; these do not appear in a derivation, so they are fitted values rather than predictions.

free parameters (4)

sigma_i2 (second sampling noise level) = 0.621 to 20.92 depending on dataset (Table 2)
Per-dataset second-step noise level experimentally determined; directly affects CoSTI-2 accuracy.
Curriculum endpoints s0, s1 = s0=10, s1=200
Chosen based on ablations in Section 5.4.6; other values perform worse on the same benchmark.
Dropout rate = 0.2
Increased from 0.1 based on observed performance improvement (Section 4.1.2).
Number of median samples = 100
Set to 100 for deterministic imputation in Section 4.1.4; directly multiplies inference cost, but not tuned.

assumptions (4)

standard math The consistency function f* exists and satisfies self-consistency along the PF-ODE (Song et al. [11]).
Basis of consistency training; cited from Song et al., assumed without proof.
standard math The PF-ODE with score function describes the noising process on multivariate time series data (Eq. 2).
Standard diffusion and consistency model theory; assumes score-based generative modeling applies to time series.
domain assumption A fixed graph topology, built from distances or correlations, captures the spatial structure of the sensors for all time steps.
Section 3.1 states a constant adjacency matrix reused across time; if spatial dependencies vary, the conditioning signal is misspecified.
domain assumption Synthetic missingness strategies during training generalize to the real missing patterns in the test sets.
Section 5.2 trains on point, block, historical, and hybrid masks; reported accuracy on real missingness depends on this transfer.

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Cite this review

Pith. "Pith review of CoSTI: Consistency Models for (a faster) Spatio-Temporal Imputation." pith.science (2026). https://pith.science/paper/PTB24QYP

@misc{pith2026250119364,
  author       = {Pith},
  title        = {Pith review of: CoSTI: Consistency Models for (a faster) Spatio-Temporal Imputation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PTB24QYP}},
  note         = {Machine review of arXiv:2501.19364}
}

read the original abstract

Multivariate Time Series Imputation (MTSI) is crucial for many applications, such as healthcare monitoring and traffic management, where incomplete data can compromise decision-making. Existing state-of-the-art methods, like Denoising Diffusion Probabilistic Models (DDPMs), achieve high imputation accuracy; however, they suffer from significant computational costs and are notably time-consuming due to their iterative nature. In this work, we propose CoSTI, an innovative adaptation of Consistency Models (CMs) for the MTSI domain. CoSTI employs Consistency Training to achieve comparable imputation quality to DDPMs while drastically reducing inference times, making it more suitable for real-time applications. We evaluate CoSTI across multiple datasets and missing data scenarios, demonstrating up to a 98% reduction in imputation time with performance on par with diffusion-based models. This work bridges the gap between efficiency and accuracy in generative imputation tasks, providing a scalable solution for handling missing data in critical spatio-temporal systems.

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