DiffImpute: Diffusion Imputation Method

• DiffImpute is a diffusion-based imputation framework that uses DDPM with mask-aware sampling to accurately reconstruct missing values.
• It incorporates techniques such as Harmonization and refined non-Markovian sampling to enhance data consistency and reduce inference cost.
• Extended variants adapt the core method for MNAR, heterogeneous data, and cross-domain settings, broadening its practical applications.

DiffImpute denotes a diffusion-model-based approach to missing-value imputation, and more specifically the DDPM-based tabular imputer introduced in “DiffImpute: Tabular Data Imputation With Denoising Diffusion Probabilistic Model” (Wen et al., 2024). In that original formulation, a denoising diffusion probabilistic model is trained on complete tabular datasets and then used to reconstruct missing entries while preserving observed values through mask-aware sampling, with additional mechanisms such as Harmonization and a refined non-Markovian sampler (Wen et al., 2024). Subsequent work uses the term in a broader sense for conditional, score-based, residual, EM-based, MNAR-aware, cross-domain, and uncertainty-aware imputers across tabular, time-series, image, spatiotemporal, and microbiome settings (Zheng et al., 2022, Zhang et al., 2024, Sim et al., 25 May 2026). This suggests a broader editorial usage in which “DiffImpute” refers not only to one named method but to a family of diffusion-based imputation systems.

1. Definition and historical placement

The original DiffImpute method addresses tabular data imputation under Missing Completely At Random (MCAR) and Missing At Random (MAR) by training a DDPM on complete rows and then conditioning the reverse process on observed entries at inference time (Wen et al., 2024). Its design premise is that a generative model of complete rows can produce credible imputations for missing entries “without undermining the authenticity of the existing data,” and that the reverse trajectory can be made more coherent through Harmonization and a refined non-Markovian sampling process (Wen et al., 2024).

A closely related but earlier tabular line is “Conditional Score-based Diffusion Models for Tabular data” (TabCSDI), which adapts CSDI-style conditional diffusion to tabular vectors and explicitly studies one-hot encoding, analog bits encoding, and feature tokenization for mixed numerical and categorical variables (Zheng et al., 2022). Whereas DiffImpute trains on complete tabular datasets and conditions at inference through mask-aware sampling, TabCSDI directly learns the conditional distribution $p_\theta(\mathbf{x}_0^{ta} \mid \mathbf{x}_0^{co})$ for target versus conditional features (Zheng et al., 2022).

Later work broadened the methodological scope. DiffPuter combines an unconditional score-based diffusion model with an Expectation-Maximization algorithm so that diffusion training corresponds to the M-step and conditional sampling to the E-step (Zhang et al., 2024). PRDIM extends diffusion imputation to Missing Not at Random (MNAR) by jointly modeling data and missing mask, $p_{\theta,\phi}(X,M)=p_\theta(X)p_\phi(M\mid X)$, through a diffusion backbone, a pattern recognizer, and an EM loop (Sim et al., 25 May 2026). MissHDD argues that a single diffusion process is insufficient for heterogeneous tabular data and separates continuous DDIM-based denoising from a discrete latent-path diffusion channel (Zhou et al., 18 Nov 2025). Diff-Joint further generalizes the problem by treating some na values as meaningfully missing and therefore not necessarily imputable, learning a joint diffusion model over data and a latent missingness mask (Zhang et al., 3 Jun 2026).

The same broad pattern appears outside conventional tabular imputation. DSDI, DiffImp, RDPI, and LSSDM use diffusion for probabilistic time-series or spatiotemporal imputation, each altering the denoiser, conditioning pathway, or latent target (Xiao et al., 11 Jan 2025, Gao et al., 2024, Liu et al., 2024, Liang et al., 2024). CD$^2$-TSI adds cross-domain adaptation, while DepMicroDiff adapts conditional diffusion to sparse microbiome abundance tables with a Dependency-Aware Transformer and LLM-conditioned metadata (Zhang et al., 14 Jun 2025, Sadia et al., 31 Jul 2025).

2. Core probabilistic formulation

Across the literature, DiffImpute-style methods share the same generative backbone: a forward diffusion process that gradually corrupts data and a learned reverse process that reconstructs missing content conditioned on observed content. In the original DiffImpute, the DDPM forward chain is

$$q(\mathbf{x}_t \mid \mathbf{x}_{t-1}) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\alpha_t}\mathbf{x}_{t-1},\, (1 - \alpha_t)\mathbf{I}\Bigr),$$

with closed form

$$q(\mathbf{x}_t \mid \mathbf{x}_0) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\bar{\alpha}_t}\,\mathbf{x}_0,\; (1 - \bar{\alpha}_t)\mathbf{I}\Bigr),$$

and a reverse model

$$p_\theta(\mathbf{x}_{t-1} \mid \mathbf{x}_t) = \mathcal{N}\bigl(\mathbf{x}_{t-1};\, \boldsymbol{\mu}_\theta(\mathbf{x}_t, t),\, \sigma_t^2 \mathbf{I}\bigr),$$

parameterized through a denoising network $f_\theta(\mathbf{x}_t,t)$ that predicts the Gaussian noise (Wen et al., 2024). The training objective is a smooth L1 noise-prediction loss, while TabCSDI uses the corresponding conditional diffusion loss for target features given conditional features (Zheng et al., 2022).

For imputation, the central object is the conditional distribution over missing entries given observed ones. In DiffImpute this is enforced procedurally: the observed region is re-noised from the actual observed data at every reverse step, the unknown region is denoised by the model, and both are merged by the binary mask (Wen et al., 2024). TabCSDI formalizes the same logic by splitting each example into a conditional part $\mathbf{x}^{co}$ and a target part $\mathbf{x}^{ta}$, learning $p_\theta(\mathbf{x}_0^{ta} \mid \mathbf{x}_0^{co})$ directly (Zheng et al., 2022).

DiffPuter makes the latent-variable structure explicit. Let $p_{\theta,\phi}(X,M)=p_\theta(X)p_\phi(M\mid X)$0. Its forward perturbation is

$p_{\theta,\phi}(X,M)=p_\theta(X)p_\phi(M\mid X)$1

and the reverse-time SDE is

$p_{\theta,\phi}(X,M)=p_\theta(X)p_\phi(M\mid X)$2

The score-matching loss upper-bounds the negative log-likelihood of the data density, which is why the diffusion training step functions as an approximate M-step, while the tailored conditional sampler functions as an E-step (Zhang et al., 2024).

PRDIM modifies this standard probabilistic picture to handle non-ignorable missingness. With $p_{\theta,\phi}(X,M)=p_\theta(X)p_\phi(M\mid X)$3 and $p_{\theta,\phi}(X,M)=p_\theta(X)p_\phi(M\mid X)$4, it maximizes the observed-data likelihood

$p_{\theta,\phi}(X,M)=p_\theta(X)p_\phi(M\mid X)$5

under an ELBO and an EM procedure (Sim et al., 25 May 2026). This changes the target from learning $p_{\theta,\phi}(X,M)=p_\theta(X)p_\phi(M\mid X)$6 alone to learning the joint $p_{\theta,\phi}(X,M)=p_\theta(X)p_\phi(M\mid X)$7, which is essential under MNAR because $p_{\theta,\phi}(X,M)=p_\theta(X)p_\phi(M\mid X)$8 depends on unobserved values (Sim et al., 25 May 2026).

Diff-Joint generalizes the same principle in a different direction. It augments the domain with an explicit na state and models a joint diffusion state $p_{\theta,\phi}(X,M)=p_\theta(X)p_\phi(M\mid X)$9, where $^2$0 is a latent mask of meaningfully missing entries. The forward SDE and denoising score matching are then applied to the concatenated representation of data and mask, so the model learns $^2$1 rather than only $^2$2 (Zhang et al., 3 Jun 2026).

3. Conditioning mechanisms and denoising architectures

A defining feature of DiffImpute systems is how conditioning on observed information is injected into the denoiser. The original DiffImpute uses a mask-aware reverse trajectory in which known entries are repeatedly replaced by noised versions of the actual observed values, while unknown entries follow the learned reverse update (Wen et al., 2024). To support this in tabular data, it introduces four tabular denoising networks—MLP, ResNet, Transformer, and a tabular U-Net—and a Time Step Tokenizer that produces learned scale-and-shift embeddings for each feature dimension (Wen et al., 2024). In its reported experiments, the Transformer denoiser is the strongest variant (Wen et al., 2024).

TabCSDI adapts CSDI from time series to feature-wise tabular vectors by removing temporal architecture and using a transformer plus MLP over features, while systematically studying one-hot encoding, analog bits encoding, and feature tokenization for mixed data types (Zheng et al., 2022). Feature tokenization is particularly relevant because it places numerical and categorical features into a common embedding space of fixed dimension, which proved especially effective for multi-category variables on Census (Zheng et al., 2022).

Time-series and spatiotemporal variants emphasize different backbones. DSDI couples a conditional DDPM-style imputer with a boundary-enhancing weight-reducing injection strategy and a multi-scale temporal S4-based U-Net, targeting long contiguous gaps and long-range dependencies (Xiao et al., 11 Jan 2025). DiffImp replaces Transformer or CNN denoisers with a Bidirectional Attention Mamba block and a Channel Mamba Block, giving linear-time sequence modeling with explicit bidirectional temporal encoding and inter-variable coupling (Gao et al., 2024). RDPI uses a two-stage structure in which a deterministic imputer first predicts a coarse reconstruction, and a conditional diffusion model then refines residuals rather than raw values; its denoiser combines 1D convolution, temporal self-attention, graph neural networks, and spatial self-attention (Liu et al., 2024).

Other variants modify the conditioning substrate itself. LSSDM first learns a latent distribution via a variational graph autoencoder, reconstructs coarse missing values, and then refines them with a conditional diffusion model in data space, effectively coupling a latent prior with diffusion refinement (Liang et al., 2024). CD$^2$3-TSI adds shared side information and domain-specific denoising networks for source and target domains, alongside FFT-based FMixup priors and output-level cross-domain consistency alignment (Zhang et al., 14 Jun 2025). DepMicroDiff places conditional diffusion in a VAE latent space and supplies context through both observed latent values and BERT-encoded patient metadata; its Dependency-Aware Transformer uses a binary dependency matrix derived from directional relations and mutual information to constrain attention among taxa (Sadia et al., 31 Jul 2025).

The original DiffImpute also introduces two sampling-time mechanisms that became conceptually influential. Harmonization repeatedly re-noises and re-denoises intermediate samples to improve coherence between observed and imputed regions, and Impute-DDIM replaces the full Markovian chain with a refined non-Markovian sampler that remains compatible with Harmonization while reducing inference cost (Wen et al., 2024).

4. Missingness regimes, latent masks, and selective imputation

One major axis of variation within DiffImpute research is the treatment of the missingness mechanism itself. The original DiffImpute is explicitly positioned for MCAR and MAR, and does not model the mask generatively (Wen et al., 2024). TabCSDI likewise focuses on conditional diffusion over missing features without an explicit missingness model (Zheng et al., 2022). DiffPuter is agnostic at the level of the diffusion objective and is evaluated under MCAR, MAR, and MNAR, but its probabilistic target remains the full-data density $^2$4 and the conditional $^2$5 derived from it (Zhang et al., 2024).

PRDIM changes this by treating MNAR as a first-class generative object. Its pattern recognizer $^2$6 parameterizes $^2$7 through a Bernoulli factorization and binary cross-entropy loss, and its gradient is injected into the reverse update: $^2$8 This means the reverse score is guided not only by data likelihood but also by mask likelihood, making the imputation consistent with the learned missing pattern under MNAR (Sim et al., 25 May 2026).

Diff-Joint addresses a different conceptual limitation: not every missing entry should be imputed. It distinguishes “meaningfully missing” entries, where na is a semantically valid state, from observation-induced missing values that should be recovered (Zhang et al., 3 Jun 2026). For each missing entry, it estimates both an imputed value and a meaningful-missingness label $^2$9. The update rule combines uncertainty clustering and majority vote over sampled latent masks: $$q(\mathbf{x}_t \mid \mathbf{x}_{t-1}) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\alpha_t}\mathbf{x}_{t-1},\, (1 - \alpha_t)\mathbf{I}\Bigr),$$0 Entries labeled as meaningful missing are preserved as na, whereas only those labeled ordinary missing are imputed (Zhang et al., 3 Jun 2026).

MissHDD reframes the problem from the perspective of feature geometry rather than missingness semantics. It argues that conditional consistency and information collapse arise because conventional Gaussian diffusion treats heterogeneous variables as though they lived on a common Euclidean manifold (Zhou et al., 18 Nov 2025). Its remedy is a continuous DDIM-based channel for numerical variables and a loopholing-based simplex-preserving channel for categorical variables. For the latter, the forward kernel is

$$q(\mathbf{x}_t \mid \mathbf{x}_{t-1}) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\alpha_t}\mathbf{x}_{t-1},\, (1 - \alpha_t)\mathbf{I}\Bigr),$$1

which keeps every categorical state on the simplex at every time step (Zhou et al., 18 Nov 2025).

Cross-domain imputation introduces yet another missingness-related complication: incompleteness hinders adaptation because the target domain is both shifted and missing. CD$$q(\mathbf{x}_t \mid \mathbf{x}_{t-1}) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\alpha_t}\mathbf{x}_{t-1},\, (1 - \alpha_t)\mathbf{I}\Bigr),$$2-TSI treats this as a joint data, model, and algorithm problem, using FMixup to fill only original missing values with low-frequency cross-domain spectral priors and CDCA to align source and target denoisers only when their outputs are moderately similar (Zhang et al., 14 Jun 2025).

5. Representative variants, empirical performance, and applications

The published record presents DiffImpute as a high-performing but heterogeneous family. The table summarizes representative variants and their reported differentiating mechanisms.

Method	Core mechanism	Setting
DiffImpute (Wen et al., 2024)	DDPM on complete tabular data; Harmonization; Impute-DDIM; MLP/ResNet/Transformer/U-Net denoisers	Tabular, MCAR/MAR
TabCSDI (Zheng et al., 2022)	Conditional score-based diffusion with one-hot, analog bits, and feature tokenization	Tabular, mixed numerical/categorical
DiffPuter (Zhang et al., 2024)	Unconditional score model with EM; M-step as diffusion training; E-step as conditional sampling	In-sample and out-of-sample tabular imputation
PRDIM (Sim et al., 25 May 2026)	Diffusion backbone plus pattern recognizer $$q(\mathbf{x}_t \mid \mathbf{x}_{t-1}) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\alpha_t}\mathbf{x}_{t-1},\, (1 - \alpha_t)\mathbf{I}\Bigr),$$3; ELBO and EM under MNAR	Time series, images, tabular
MissHDD (Zhou et al., 18 Nov 2025)	Continuous DDIM channel plus loopholing discrete diffusion channel	Heterogeneous tabular, MCAR/MAR/MNAR
Diff-Joint (Zhang et al., 3 Jun 2026)	Joint diffusion over data and latent meaningful-missingness mask	Selective tabular imputation

The original DiffImpute reports that, “when paired with the Transformer as the denoising network, it consistently outperforms its competitors, boasting an average ranking of 1.7 and the most minimal standard deviation,” while “the next best method lags with a ranking of 2.8 and a standard deviation of 0.9” (Wen et al., 2024). DiffPuter reports, in its abstract, “an average improvement of 6.94% in MAE and 4.78% in RMSE compared to the most competitive existing method” across ten diverse datasets (Zhang et al., 2024). MissHDD reports higher imputation accuracy, more stable sampling trajectories, and improved robustness across MCAR, MAR, and MNAR compared with existing diffusion-based and classical methods (Zhou et al., 18 Nov 2025).

Time-series and spatiotemporal variants also report strong gains. DSDI states that it outperforms existing imputation methods and attributes the gains to boundary-aware AR injection and a multi-scale S4-based U-Net; for example, on ETT Point MAE it reports $$q(\mathbf{x}_t \mid \mathbf{x}_{t-1}) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\alpha_t}\mathbf{x}_{t-1},\, (1 - \alpha_t)\mathbf{I}\Bigr),$$4 versus $$q(\mathbf{x}_t \mid \mathbf{x}_{t-1}) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\alpha_t}\mathbf{x}_{t-1},\, (1 - \alpha_t)\mathbf{I}\Bigr),$$5 for MTSCI, and on AQI Simulated MAE $$q(\mathbf{x}_t \mid \mathbf{x}_{t-1}) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\alpha_t}\mathbf{x}_{t-1},\, (1 - \alpha_t)\mathbf{I}\Bigr),$$6 versus $$q(\mathbf{x}_t \mid \mathbf{x}_{t-1}) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\alpha_t}\mathbf{x}_{t-1},\, (1 - \alpha_t)\mathbf{I}\Bigr),$$7 for MTSCI (Xiao et al., 11 Jan 2025). DiffImp reports substantially improved probabilistic calibration on Electricity, with CRPS-sum $$q(\mathbf{x}_t \mid \mathbf{x}_{t-1}) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\alpha_t}\mathbf{x}_{t-1},\, (1 - \alpha_t)\mathbf{I}\Bigr),$$8 versus $$q(\mathbf{x}_t \mid \mathbf{x}_{t-1}) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\alpha_t}\mathbf{x}_{t-1},\, (1 - \alpha_t)\mathbf{I}\Bigr),$$9 for SSSD and $$q(\mathbf{x}_t \mid \mathbf{x}_0) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\bar{\alpha}_t}\,\mathbf{x}_0,\; (1 - \bar{\alpha}_t)\mathbf{I}\Bigr),$$0 for CSDI, and faster sampling than both CSDI and SSSD on the reported benchmarks (Gao et al., 2024). RDPI reports reduced sampling cost and strong MSE improvements on air-quality datasets; on AQI36 in-sample it reports MAE 7.98 and MSE 238.25, compared with MAE 9.41 and MSE 361.28 for MIDM and MAE 9.60 and MSE 372.49 for CSDI (Liu et al., 2024).

The family has also expanded into modality-specific applications. LSSDM reports best MAE and CRPS among the compared generative baselines on AQI-36, PhysioNet 2012, and PeMS-BAY, coupling a variational graph autoencoder with conditional diffusion refinement (Liang et al., 2024). DepMicroDiff reports microbiome reconstruction gains on TCGA cancer datasets, reaching Pearson correlation up to 0.712 and cosine similarity up to 0.812 while lowering RMSE and MAE across multiple cancer types (Sadia et al., 31 Jul 2025). ImDiffusion uses diffusion-based time-series imputation as an anomaly-detection primitive and reports a 11.4% increase in detection F1 score in a Microsoft production system compared to the legacy approach (Chen et al., 2023).

Several papers emphasize that improvements are especially pronounced when the modeling assumption matches the pathology of the data. PRDIM reports that its gains are particularly large on real MNAR masks and smaller under MAR or MCAR, where the pattern recognizer becomes less informative (Sim et al., 25 May 2026). Diff-Joint reports substantially better token-level recovery accuracy and downstream task performance when missingness itself carries signal, precisely because it can preserve semantically meaningful na values instead of forcing a regular-value reconstruction (Zhang et al., 3 Jun 2026).

6. Limitations and open directions

Despite strong empirical results, the literature identifies persistent limitations. The original DiffImpute requires complete training data, treats all features as continuous after preprocessing, and still incurs a substantial sampling cost even with Impute-DDIM (Wen et al., 2024). DiffPuter notes that each EM iteration retrains or further trains a diffusion model, which is more expensive than shallow predictive models or simple statistical imputers, and it does not prove global convergence of the outer EM loop (Zhang et al., 2024).

Categorical and mixed-type data remain a recurring difficulty. TabCSDI shows that categorical embedding choices matter substantially, especially for multi-category features, implying that naïve continuous relaxations are often inadequate (Zheng et al., 2022). MissHDD makes this criticism explicit, arguing that Gaussian diffusion over heterogeneous tables produces semantic drift for categorical variables and instability for numerical ones, and that structure-aware dual-channel diffusion is therefore necessary (Zhou et al., 18 Nov 2025). PRDIM notes a different modeling limitation: its pattern recognizer assumes conditionally independent Bernoulli indicators given $$q(\mathbf{x}_t \mid \mathbf{x}_0) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\bar{\alpha}_t}\,\mathbf{x}_0,\; (1 - \bar{\alpha}_t)\mathbf{I}\Bigr),$$1, while real missingness can be block-level or hierarchical (Sim et al., 25 May 2026).

Computation is another common bottleneck. PRDIM requires gradients of the pattern recognizer loss with respect to the diffusion state at every reverse step, which “roughly doubles or more the cost vs unguided diffusion” (Sim et al., 25 May 2026). DSDI, DiffImp, RDPI, and DepMicroDiff all retain the generic diffusion burden of repeated denoising steps, even when specialized samplers or efficient backbones reduce that burden (Xiao et al., 11 Jan 2025, Gao et al., 2024, Liu et al., 2024, Sadia et al., 31 Jul 2025).

Several open directions recur across papers. One is explicit treatment of the missingness mechanism: PRDIM addresses MNAR, but most DiffImpute variants still assume ignorable or unspecified missingness (Sim et al., 25 May 2026). Another is selective imputation: Diff-Joint argues that some missing entries should remain missing, not be filled, and operationalizes that distinction through joint diffusion over data and latent mask (Zhang et al., 3 Jun 2026). A third is domain and modality adaptation: CD$$q(\mathbf{x}_t \mid \mathbf{x}_0) = \mathcal{N}\Bigl(\mathbf{x}_t;\, \sqrt{\bar{\alpha}_t}\,\mathbf{x}_0,\; (1 - \bar{\alpha}_t)\mathbf{I}\Bigr),$$2-TSI extends diffusion imputation to domain shift, while DepMicroDiff uses LLM-conditioned metadata and dependency-aware attention to incorporate structured context (Zhang et al., 14 Jun 2025, Sadia et al., 31 Jul 2025).

Taken together, these developments indicate that DiffImpute has evolved from a DDPM-based tabular imputer into a broader research program centered on conditional generative reconstruction under incomplete observation. The main unresolved questions concern how best to combine calibrated uncertainty, efficient sampling, explicit missingness modeling, heterogeneous variable geometry, and semantically meaningful absence within a unified imputation framework.