GAUDI: Geometry-Aware Diffusion for Calibrated Air-Quality Time-Series Imputation

1 Introduction

• Contiguous gaps can remove same-variable local context, making simultaneously observed variables more important for reconstruction.Missingness geometry refers to the arrangement of missing entries across the time–variable grid.Source: §1 · p. 1
• The design restricts auxiliary positional conditioning without removing temporal operators or changing how the network represents sequence structure.Whether this restriction improves performance is an empirical question, not a consequence of supplying fewer inputs.Source: §1 · p. 1
• Feature-side conditioning suppresses within-window position embeddings while retaining visible values, the observation mask, variable identity, and the shared temporal and feature modeling architecture.The matched variants keep architecture, input dimensions, objective, and sampling procedure fixed, isolating conditioning choice.Source: §1 · pp. 1–2
• Mean RMSE fell from 0.4792 to 0.4524 versus full-side conditioning in the expanded ItalyAir evaluation, a 5.60% improvement across all three matched seeds.The archived experiment separately reports RMSE 0.340 ± 0.020 for feature-side conditioning, versus 0.355 ± 0.006 for full context and 0.355 ± 0.020 for local CSDI.Source: §1 · p. 2
• The study also compares variable-identity ablations and examines empirical distribution scores, coverage, variable-level errors, and chronological cases.These analyses distinguish reconstruction accuracy from uncertainty calibration; they do not introduce a calibration procedure.Source: §1 · p. 2

Dependency modeling for imputation

• BRITS, SAITS, and GRU-D model temporal or cross-variable structure, with GRU-D also incorporating masks and elapsed time to distinguish unavailable from observed measurements.The paper positions its question as complementary: which auxiliary embeddings to expose in probabilistic imputation.Source: §2 · p. 2

Probabilistic and diffusion-based imputation

• Diffusion imputation includes CSDI conditioned on observed entries and SSSD with structured state-space modeling; GAUDI studies selective side conditioning rather than a new objective or sampler.The cited related approaches also include adversarial GAIN and Gaussian-process-prior GP-VAE.Source: §2 · p. 2

Structured missingness and conditioning design

• Prior structured-missingness work addresses consecutive gaps, while this study changes position and variable embeddings independently of the observation mask and temporal operators.The paper frames this as a narrower conditioning comparison than an architectural ablation or cross-protocol comparison.Source: §2 · p. 2

Observation sufficiency beyond imputation

• Related studies consider inference from observation windows and incomplete market signals, whereas this paper asks which side information diffusion imputation should use under contiguous missingness.These neighboring applications concern different downstream tasks.Source: §2 · pp. 2–3

3.1 Problem Formulation

• The formulation separates visible conditioning entries from supervised targets, evaluates only hidden entries with ground truth, and uses offline windows where observations on either side may provide context.Native missing entries use zero placeholders but are not supervised targets.Source: §3.1 · p. 3

3.2 Mask and Feature Conditioning

• GAUDI suppresses within-window position embeddings while retaining variable identity, visible values, the observation mask, sequence order, diffusion-step embeddings, and temporal processing.The compared configurations use full-side conditioning (1, 1), feature-side conditioning (0, 1), or no-side conditioning (0, 0); all retain the mask and visible-value channels.Source: §3.2 · p. 3

3.3 Temporal–Feature Denoising

• The shared denoiser combines state-space sequence processing with temporal attention across time per variable and feature attention across variables per time step.Axis-specific attention and side-tensor conditioning are shared across variants; the evaluated contribution is the conditioning policy, not a new attention or gating primitive.Source: §3.3 · p. 3

3.4 Learning and Sampling

• Training separates noisy unavailable values from clean conditioning channels and uses no auxiliary reconstruction loss.The reverse process starts from Gaussian noise, supplies visible values at every step, and is shared across variants.Source: §3.4 · p. 4
• The evaluation draws 20 samples over 50 reverse steps, uses the entrywise lower median as its point estimate, and evaluates empirical-quantile interval coverage and width separately from point error.Completed windows preserve observed entries, and the expanded evaluation applies no post-hoc calibration transformation.Source: §3.4 · p. 4

4.1 Archived Block-Missing Experiment

• RMSE 0.340 ± 0.020 for archived feature-side conditioning, versus 0.355 ± 0.006 for full context and 0.355 ± 0.020 for local CSDI.These values are reproduced from archived runs and are not recomputed from the expanded evaluation.Source: §4.1 · p. 4, Table 1 · p. 4

4.2 Expanded Evaluation Setup

• The expanded evaluation uses ItalyAir length-32 windows and 50% block-structured artificial missingness, with the same saved masks and targets across methods.The dataset has 13 measurement variables; metrics are summarized across three independently trained seeds.Source: §4.2 · pp. 4–5

4.3 Expanded Reconstruction Results

• RMSE 0.4524 and MAE 0.3131; feature-side conditioning reduces RMSE by 5.60% and MAE by 3.91% versus Full-side in the expanded evaluation.CSDI achieves the strongest aggregate performance, with RMSE 0.4127 and MAE 0.2612.Source: Table 2 · p. 4, §4.3 · p. 5

4.4 Side-Information Ablation

• GAUDI improves both RMSE and MAE over Full-side for all three matched seeds, with RMSE differences ranging from −0.0399 to −0.0165.Against No-side, GAUDI has 2.51% lower mean RMSE but slightly higher mean MAE, 0.3131 versus 0.3122.Source: §4.4 · p. 5, Table 3 · p. 5

4.5 Probabilistic Performance and Calibration

• GAUDI lowers eCRPS from 0.2558 to 0.2470 versus Full-side, but its nominal 90% interval covers only 52.89% of targets.All four methods substantially under-cover; the tested conditioning change improves reconstruction relative to Full-side without resolving predictive undercoverage.Source: §4.5 · pp. 5, 7, Table 4 · p. 6

4.6 Variable-Level Behavior

• GAUDI improves over Full-side on nine of twelve evaluable variables, but errors are not uniform across channels.It improves over CSDI on four named channels, while CSDI remains stronger on most others and in aggregate; linear interpolation performs best on absolute humidity.Source: Table 5 · p. 6, §4.6 · p. 7

4.7 Qualitative Reconstruction

• The CO(GT) example uses the chronologically selected middle test window, with predictions shown only at artificially hidden targets and observed values retained as context.The example includes close reconstructions and deviations around hidden signal changes; it is not selected by relative method performance.Source: Figure 1 · p. 6, §4.7 · p. 7

• Mean RMSE fell 5.60% relative to full-side conditioning in expanded ItalyAir, with improvement in all three matched seeds.Source: §5 · p. 7
• Feature-side conditioning separates observation geometry and variable identity from auxiliary within-window coordinates while preserving state-space and temporal–feature modeling.Source: §5 · p. 7
• The evaluation covers one dataset, one window length, and one block-structured mask suite; broader tests are needed to establish transferability.Source: §5 · p. 7