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ProCTI: Prototype-Refined Global Conditioning for Diffusion-Based Time Series Imputation
Fariza Rashid, Duc Van Le, Rahat Masood, Gustavo Batista, Aruna Seneviratne, Suranga Seneviratne
TL;DR
Diffusion-based imputation typically conditions on local context, leaving dataset-level structure implicit when observations are sparse or noisy. ProCTI retrieves global context through learned prototypes and combines it with local signals during reverse denoising. Across five datasets, it achieves the best mean MAE under random missingness, with approximately 3–30% reductions over FGTI, and is best or second-best in most attribute-wise scenarios.
Problem
Existing diffusion imputers typically use local context, while broader dataset-level patterns remain implicit and may be difficult to reflect when local observations are sparse or noisy.
Source: §1 · p. 1Method
ProCTI uses a learnable prototype bank to retrieve window-specific global context and integrate it with local signals throughout reverse diffusion.
Source: §1 · pp. 1–2Results
ProCTI achieves the best mean MAE on all five datasets under random missingness, reducing MAE by approximately 3–30% over FGTI, and is best or second-best in most attribute-wise scenarios.
Source: §1 · p. 2Takeaways & Limitations
A latent-regime analysis characterizes conditions under which prototype-derived global conditioning provably improves imputation, alongside a theoretical analysis of local-global conditioning.
Source: §1 · p. 2, §3.2.2 · p. 6The theoretical framing treats global structure as an unobserved latent regime approximated by prototypes, while local context derives from current-window observations and missingness patterns.
Source: §3.2.2 · p. 5
Abstract
from arXiv · showhide
Time series imputation has progressed from statistical and deep learning approaches to diffusion-based models, which have shown strong recent performance. Existing diffusion-based methods typically condition the reverse process using local contextual information from the current or neighbouring windows. Meanwhile, global dataset-level structure often remains implicit, limiting performance when local observations are sparse, noisy, or unrepresentative. To address this issue, we propose ProCTI, a diffusion-imputation framework that augments local conditioning with retrieved global dataset-level priors through learned prototypes. A hybrid conditioning mechanism integrates this global context with local signals during reverse diffusion, enabling more accurate reconstruction under varying missingness scenarios. Experiments across multiple benchmark datasets show that ProCTI outperforms strong baselines overall under random missingness, while remaining competitive under attribute-wise missingness. Furthermore, we use a latent-regime data model to characterise the precise conditions under which prototype-derived global conditioning provably improves imputation. We support this with a general theoretical analysis of local-global conditioning.
1 Introduction
Diffusion imputation commonly relies on local context, while dataset-level patterns may be hard to infer when observations are sparse or noisy. ProCTI retrieves global context through learned prototypes and combines it with local signals during reverse denoising.
1 Introduction
- A learnable prototype bank retrieves window-specific global context that refines conditioning throughout reverse diffusion.The retrieved context represents recurring dataset-level patterns, or regimes, and is repeatedly incorporated during denoising.Source: §1 · p. 2
- Beijing windows form four coarse PCA-space K-means groups with distinct pollution and meteorological profiles, which may be difficult to infer from a partially observed window.In a masked NO2 example, ProCTI achieves lower MAE than competing baselines by leveraging global context.Source: §1 · p. 2, Figure 1 · p. 2
- ProCTI achieves the best mean MAE on all five datasets under random missingness, reducing MAE by approximately 3–30% over the strongest diffusion baseline, FGTI.It is best or second-best in most attribute-wise missingness scenarios.Source: §1 · p. 2
- The paper introduces a latent-regime theoretical framework to characterize when prototype-derived global conditioning provably improves imputation.Its evaluation compares ProCTI with nine baselines across five real-world datasets.Source: §1 · p. 2
2 Related Work
Related work spans statistical, machine-learning, and diffusion-based time-series models, while existing diffusion imputers use varied forms of contextual conditioning. ProCTI distinguishes local window information from shared dataset-level structure and combines both for conditional imputation.
2 Related Work
- Earlier time-series methods include statistical and autoregressive approaches, matrix factorisation in TIDER, and bidirectional RNNs or self-attention in BRITS [11] and SAITS [21].Forecasting and generation work also uses recursive convolution, frequency-domain multilayer perceptrons, attention-based architectures, and diffusion models [22] [23] [24] [25] [27] [28] [29] [30].Source: §2 · pp. 2–3
- Diffusion imputers condition on observed values and dependencies in CSDI [15], spatial priors in PriSTI, neighbouring windows in MTSCI [16], or frequency priors in FGTI.TG-MSFM is a non-diffusion approach that reconciles global trends with local details during deterministic reconstruction.Source: §2 · p. 3
- ProCTI formalizes local context from the partially observed window and global context as recurring dataset-level structure shared across samples, then retrieves the latter through prototypes.Unlike PaD-TS [30], which preserves population-level statistics for unconditional generation, ProCTI uses the contexts for conditional imputation.Source: §2 · p. 3
3 Prototype-Refined Global Conditioning
ProCTI combines local window evidence with global dataset-level priors retrieved from learned prototypes during reverse diffusion. Its theoretical analysis characterizes when informative global regimes can reduce uncertainty and optimal denoising error.
3.1 Prototype-conditioning module
- A learnable prototype bank and cross-attention turn each partially observed window into an attribute-wise global context vector for conditioning reverse diffusion.The window query captures observed values and missingness patterns, while the prototypes approximate recurring temporal regimes.Source: §3.1 · pp. 3–4
3.2.1 Conditional diffusion formulation
- The conditional reverse diffusion process uses a signal derived from the partially observed window, which ProCTI refines with prototype-derived global information.The standard DDPM forward process gradually corrupts the clean input, and its denoiser predicts the injected noise.Source: §3.2.1 · pp. 4–5
3.2 Global-conditioned diffusion model
- ProCTI injects the prototype-derived regime vector into the dominant-frequency branch, combining global correction with local frequency components to form the conditioning signal.A learnable scalar controls the strength of the global correction, and the resulting signal is supplied to the denoiser at each diffusion step.Source: §3.2.2 · p. 5
3.2.2 Theoretical perspective on local and global conditioning
- Under hybrid conditioning, the conditional score separates into a local score and regime guidance, and informative guidance can reduce the Bayes-optimal denoiser’s mean-squared error.The score decomposition identifies the local term as conditioning on O and the regime term as pulling the noisy state toward states compatible with Z.Source: §3.2.2 · p. 6
- Prototype-derived global conditioning strictly improves the optimal denoiser only when the latent regime carries information about missing values beyond the observed window; otherwise it cannot improve that denoiser.Under the latent-regime model, this condition is equivalent to a lower optimal prediction error using (O, Z) than using O alone.Source: §3.2.2 · p. 6
- When the regime contains residual information about missing values, a sufficiently expressive attention map and prototype bank of size at least M⋆ can make the retrieved vector a soft sufficient statistic for the regime.If missing values are already well-approximated from observed values alone, global conditioning may provide no improvement.Source: §3.2.2 · p. 7
4 Experiments
Experiments on five real-world datasets evaluate ProCTI under stochastic segment masking and structured attribute-wise missingness, alongside component ablations. Results favor ProCTI overall, with attribute-wise performance varying by dataset and feature redundancy.
4.1 Experimental Setup
- Experiments compare ProCTI with imputation, forecasting, and diffusion-generation baselines on five real-world datasets, reporting MAE and RMSE.The datasets cover clinical, air-quality, weather, stock-price, and gait time series.Source: §4 · p. 7, §4.1 · p. 7
4.2 Imputation effectiveness
- Evaluation uses a two-state Markov segment-masking process, with training missing ratio r = 0.15 and evaluation ratios of 0.10, 0.30, 0.50, and 0.70.The average masked-segment length is 6 for both training and evaluation.Source: §4.2 · p. 8
- Approximately 3–30%: ProCTI reduces average MAE versus FGTI across stochastic segment-masking scenarios.FGTI is the second-best baseline by mean MAE in all scenarios; ProCTI's largest gains are on PhysioNet and Stock.Source: §4.2 · p. 8
4.3 Robustness under attribute-wise missingness
- The attribute-wise protocols fully mask one or two randomly selected test-window attributes and reconstruct them from remaining observed variables.PhysioNet includes only attributes with more than 80% observed values; other datasets use all available attributes.Source: §4.3 · p. 8
- Under attribute-wise masking, ProCTI has the lowest MAE and RMSE in most scenarios on Gait, PhysioNet, and Beijing, while BRITS leads on Weather and Stock.The paper links BRITS's advantage on Weather and Stock to strong cross-feature redundancy; it relates ProCTI's advantage on other datasets to informative regime structure beyond observed entries.Source: §4.3 · pp. 8–9, Table 2 · p. 8
4.4 Ablation analysis
- Across all five datasets, a learnable prototype bank improves on FGTI, and prototype cross-attention adds gains on most datasets.A learnable scaling parameter injected into the dominant-frequency branch gives the best results in most cases; PhysioNet is an exception where high-frequency-branch injection has lower MAE and RMSE.Source: §4.4 · p. 10, Table 3 · p. 10
5 Conclusion
The conclusion presents ProCTI as a prototype-retrieval diffusion imputation framework, with benefits that depend on informative regime structure and future work on interpretability and non-stationarity.
5 Conclusion
- ProCTI combines local observations with reusable global priors retrieved through prototypes to condition reverse diffusion.Its learnable prototype-conditioning module supplies window-level global context to augment reverse-diffusion conditioning signals.Source: §5 · p. 10
- Theoretical analysis and experiments support improved imputation when regimes contain information beyond observed entries.When Y ⊥⊥ Z | O holds approximately under strong cross-feature redundancy, simpler regression baselines remain competitive.Source: §5 · p. 10
- Future work includes improving prototype-bank interpretability and designing for non-stationary regimes.Source: §5 · p. 10
A.1 Related work
Prior time-series work uses prototypes for several purposes, while ProCTI applies learned dataset-level regimes to conditional diffusion imputation of general multivariate series.
A.1 Related work
- Prototype methods in anomaly detection, healthcare imputation, and generation target regular patterns, inter-series memory, or sequence synthesis, respectively.ProtoAD [37] characterises regular latent patterns; PRIME captures inter-series information from similar patients; TimeDP [39], OATS [40], and K-ProtoDiff use prototypes for generation.Source: §A.1 · p. 14
- ProCTI learns reusable dataset-level regimes for conditional imputation of general multivariate time series.Source: §A.1 · p. 14
- Unlike prior prototype approaches focused on retrieval-style memory, interpretability, or unconditional generation, ProCTI combines global prototypes with local observations during reverse diffusion.Source: §A.1 · p. 14
A.2.1 Multi-head cross-attention details
The appendix outlines a multi-head cross-attention computation, including per-head projections, context vectors, and a projected regime vector.
A.2.1 Multi-head cross-attention details
- The computation uses H attention heads, each with dimension dh = D/H and learnable projections.Source: §A.2.1 · p. 14
- Each head has projected query, key, and value vectors and yields a per-head context vector.Source: §A.2.1 · p. 14
- The regime vector is projected as rb = Wrcb ∈ K.Source: §A.2.1 · p. 14
A.2.2 Training and imputation algorithms
ProCTI trains a diffusion denoiser with local frequency features and retrieved prototype context, then uses those signals during reverse diffusion to impute missing values.
A.2.2 Training and imputation algorithms
- Training takes a time series, keep-mask, and prototype bank, and updates the denoiser and prototype module using a diffusion noise-prediction loss.The training algorithm samples a diffusion step and Gaussian noise before computing the loss.Source: §A.2.2 · p. 14
- For each incomplete window, ProCTI filters observed context and queries a learned prototype bank to form global conditioning.Cross-attention retrieves prototype context, which is transformed into a regime vector for conditioning.Source: §A.2.2 · pp. 14–15
- Imputation starts from Gaussian noise, repeatedly denoises using observed values and conditioning signals, then restores the observed entries.The final output combines the original observed values with the imputed values at masked positions.Source: §A.2.2 · pp. 14–15
A.3 Dataset preparation
Dataset preparation keeps entities separate across splits for three datasets and uses chronological splits for Weather and Stock before creating fixed-length windows.
A.3 Dataset preparation
- Gait, PhysioNet, and Beijing are split by entity to prevent information leakage and assess generalisation across non-overlapping entities.The entities are users, patients, and monitoring stations, respectively.Source: §A.3 · p. 15
- Gait and PhysioNet use entity-level 70:15:15 splits, while Beijing uses an 8:2:2 split across its 12 monitoring stations.Gait is user-disjoint, PhysioNet patient-wise, and Beijing station-disjoint.Source: §A.3 · p. 15
- Weather and Stock use chronological 70:15:15 splits, and all datasets are windowed after splitting into non-overlapping fixed-length sequences.Source: §A.3 · p. 15
A.4 Supplementary experiments
Supplementary evaluations report imputation, probabilistic reliability, resource use, hyperparameter sensitivity, prototype utilisation, visualisations, and broader-impact considerations.
A.4.1 Probabilistic evaluation
- ProCTI obtains the best or second-best CRPS across missingness ratios while keeping PI90 coverage close to 90%.Some baselines have narrow intervals but severe under-coverage; ProCTI is described as offering a more balanced trade-off among accuracy, calibration, and sharpness.Source: Table 4 · p. 15, §A.4.1 · p. 16, Table 5 · p. 16
A.4.2 Resource consumption
- ProCTI runs comparably to FGTI and faster than CSDI and MTSCI, while using GPU memory similar to FGTI and more than most baselines.Source: §A.4.2 · p. 16, Figure 4 · p. 16
A.4.3 Hyperparameter evaluation
- 6/10 MAE and RMSE cells favor a prototype bank size of 32; α = 0.1 performs best overall, though α = 0.3 has lower MAE on PhysioNet and Weather.α = 0.1 wins on Beijing, Gait, and Stock.Source: §A.4.3 · p. 17, Table 6 · p. 17, Figure 5 · p. 17
A.5.1 Imputation effectiveness
- ProCTI achieves the lowest errors in the majority of evaluated Markov-masking scenarios, with FGTI usually second best.Table 7 reports MAE and RMSE at each missingness ratio, averaged over five repeated runs.Source: §A.5.1 · p. 17, Table 7 · p. 18
A.5.2 Statistical significance of imputation effectiveness
- On Beijing, ProCTI significantly outperforms FGTI (p < 0.05) in 4/8 cells by t-test and 1/8 by Wilcoxon; on Gait, it is significantly better in 5/8 cells under both tests.Source: §A.5.2 · p. 17
A.5.3 Prototype utilisation
- At least 31 of 32 prototypes receive more than half a uniform share of attention, and 27–29 are needed to account for 90% of attention mass.Stock and Gait need 27 and 28 prototypes, respectively, while Weather, PhysioNet, and Beijing each need 29.Source: §A.5.3 · p. 19, Table 9 · p. 19
A.6 Visualisations
- Across Beijing, Gait, and PhysioNet examples, ProCTI follows masked ground truth more closely than CSDI, while often looking similar to FGTI.Figure 9 also visualises Stock windows, prototype attention, regime-vector values, and dominant-frequency adjustments.Source: §A.6 · p. 19, Figure 6 · p. 20, Figure 7 · p. 21
A.7 Broader impact statement
- The authors recommend using ProCTI as decision support rather than a replacement for human judgment, especially in high-stakes settings.Incorrect imputations may have unintended downstream consequences.Source: §A.7 · p. 19