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

Xinjin Li; Yudi Xia; Calvin Chang Liu; Weiru Lin; Bojun Li; Ziwei Hong; Bolun Zhang; Jinghan Cao

cs.LGSep 24, 2026

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

Authors: Xinjin Li, Yudi Xia, Calvin Chang Liu, Weiru Lin, Bojun Li, Ziwei Hong, Bolun Zhang, Jinghan Cao,

Organizations: Columbia University · Independent Researcher · University of California, Davis · University of California, San Diego ·

Abstract

Air-quality sensor outages often create contiguous missing blocks, where side information useful for isolated missingness may be less reliable. We study a block-specific, GAUDI-aligned conditional diffusion imputer that retains temporal and feature processing, visible-value and mask conditioning, variable identity, and diffusion-step information, while suppressing absolute time-position side embeddings. On ItalyAir (13 variables, length-32 windows, nominal 50% block missingness; three archived seeds), this feature-side configuration achieves RMSE 0.340, versus 0.355 for full context and 0.355 for local CSDI. The experiment isolates a geometry-aware conditioning effect under block missingness.

Figures & tables

Method	RMSE $\downarrow$
TCN-diff (local)	$0.834\pm 0.010$
DilatedConv-diff (local)	$0.693\pm 0.007$
CSDI (local adaptation)	$0.355\pm 0.020$
Full-context counterpart	$0.355\pm 0.006$
Block-specific feature-side	$\mathbf{0.340\pm 0.020}$

Table 1: Archived ItalyAir experiment reported in the ICONIP extended abstract. RMSE mean

\pm

SD over seeds 1000–1002; values are reproduced from that report, not recomputed from the expanded evaluation. Bold marks the lowest reported mean.

Method	RMSE $\downarrow$	MAE $\downarrow$
Mean	$1.5315$	$1.2120$
Linear interpolation	$0.7720$	$0.4862$
CSDI (adapted)	$\mathbf{0.4127\,{\pm}\,0.0022}$	$\mathbf{0.2612\,{\pm}\,0.0018}$
Full-side	$0.4792\,{\pm}\,0.0187$	$0.3259\,{\pm}\,0.0083$
No-side	$0.4641\,{\pm}\,0.0111$	$0.3122\,{\pm}\,0.0066$
GAUDI (feature-only)	$0.4524\,{\pm}\,0.0148$	$0.3131\,{\pm}\,0.0131$

Table 2: Expanded evaluation: reconstruction on ItalyAir with 50% block-structured artificial missingness. Neural results are mean

\pm

sample SD over three training seeds; deterministic methods have no seed SD. All rows use the same targets. Bold marks the lowest mean in each column.

Comparator	Seed	$\Delta$ RMSE	$\Delta$ MAE
Full-side	2101	$-0.0399$	$-0.0167$
	2102	$-0.0165$	$-0.0159$
	2103	$-0.0241$	$-0.0057$
No-side	2101	$-0.0274$	$-0.0079$
	2102	$-0.0106$	$-0.0069$
	2103	$+0.0030$	$+0.0177$

Table 3: Expanded evaluation: matched-seed effects of feature-side conditioning. Differences are GAUDI minus the named comparator; negative values favor GAUDI. All listed configurations share the backbone, training budget, and sampler.

Method	eCRPS $\downarrow$	Coverage 80 (%)	Width 80	Coverage 90 (%)	Width 90
CSDI (adapted)	$\mathbf{0.1984\,{\pm}\,0.0018}$	$53.82\,{\pm}\,1.20$	$0.4648\,{\pm}\,0.0050$	$63.03\,{\pm}\,0.95$	$0.5779\,{\pm}\,0.0056$
Full-side	$0.2558\,{\pm}\,0.0071$	$45.13\,{\pm}\,0.83$	$0.4730\,{\pm}\,0.0047$	$53.52\,{\pm}\,0.95$	$0.5879\,{\pm}\,0.0072$
No-side	$0.2421\,{\pm}\,0.0057$	$51.10\,{\pm}\,1.02$	$0.5367\,{\pm}\,0.0109$	$59.83\,{\pm}\,0.86$	$0.6723\,{\pm}\,0.0134$
GAUDI (feature-only)	$0.2470\,{\pm}\,0.0120$	$44.81\,{\pm}\,2.15$	$0.4639\,{\pm}\,0.0044$	$52.89\,{\pm}\,1.99$	$0.5789\,{\pm}\,0.0074$

Table 4: Expanded evaluation: distributional performance from the same 20 samples per run. eCRPS and interval widths use standardized units; coverage is reported as a percentage. Values are mean

\pm

sample SD over three seeds. Lower eCRPS is better; coverage is compared with the nominal 80% and 90% levels, and width is interpreted jointly with coverage.

Variable	$N_{d}$	Mean	Linear	CSDI	Full-side	No-side	GAUDI
CO(GT)	575	$1.0055$	$0.7165$	$0.4201$	$0.3676$	$\mathbf{0.3473}$	$0.3581$
PT08.S1(CO)	810	$0.9616$	$0.5487$	$\mathbf{0.2455}$	$0.2778$	$0.2838$	$0.2611$
NMHC(GT)	0	—	—	—	—	—	—
C6H6(GT)	1,024	$0.9829$	$0.6667$	$\mathbf{0.2454}$	$0.2707$	$0.2798$	$0.2772$
PT08.S2(NMHC)	1,043	$1.1086$	$0.7365$	$\mathbf{0.2563}$	$0.2822$	$0.2852$	$0.2798$
NOx(GT)	976	$2.0581$	$1.2259$	$0.6464$	$0.7111$	$0.7189$	$\mathbf{0.6306}$

Table 5: Expanded evaluation: per-variable RMSE, averaged over training seeds for neural methods.

N_{d}

is the number of evaluated cells. All 13 input variables are retained; — denotes no evaluable target. Bold marks the lowest mean in each evaluable row.

Figure 1: Expanded evaluation: CO(GT) reconstruction in the chronologically selected middle test window (ordinal 29). Curves and markers use the seed-2101 models, not an ensemble across training seeds. Predictions are shown only at artificially hidden targets; observed values remain available as context. The displayed window is selected chronologically rather than by relative method performance.

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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.

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