Iterative Missing Data Imputation with Model Form Adaptation and Non-Missing Feature Supervision
Authors: Hao Wang
zhengnan li
Zhichao Chen
Xu Chen
Shuting He
Guangyi Liu
Haoxuan Li
Zhouchen Lin
Venue: NeurIPS 2025 poster ()
Keywords: missing data imputation
missing data completion
missing value imputation
kernel
ridge regression
non-missing feature
Primary area: applications

Abstract: Iterative imputation is a prevalent method for missing data imputation, where each feature is imputed iteratively by treating it as a target variable estimated from all other features. However, iterative imputation method suffers from two principal limitations: 
(1) it imposes a single parametric model form to impute all features, neglecting the potential for optimal models to vary among features, which risks model misspecification; and
(2) it assumes every feature contains missing values, overlooking the potential presence of non-missing features, termed as oracle features, which are informative for imputation. 
To address these limitations, we propose kernel point imputation (KPI), a bi-level optimization framework for iterative missing data imputation. 
At the inner level, KPI adaptively learns the optimal model form for each feature within a reproducing kernel Hilbert space, addressing limitation (1). At the outer level, KPI utilizes oracle features as supervisory signals to iteratively refine the imputations, addressing limitation (2). 
Experiments demonstrate that KPI outperforms competitive imputation methods. Code is available at https://github.com/FMLYD/kpi.git.

Iterative Missing Data Imputation with Model Form Adaptation and Non-Missing Feature Supervision
Hao Wang 1M Zhengnan Li
Xu Chen Shuting He* Haoxuan Li Zhouchen Lin 2:7,8
Zhichao Chen
Guangyi Liu
Xiiohhnngshu Inc.
State Key Lab of General AI, School of Intelligence Science and Technology, Peking University
Gaoling School of Artificial Intelligence, Renmin University of China "School of Computing and Artificial Intelligence, Shanghai University of Finance and Economics Department of Control Science and Engineering. Zhejiang University
"Center for Data Science, Peking University Institute for Artificial Intelligence, Peking University
Pazhou Laboratory (Huangpu), Guangzhou, Guangdong, China
Abstract
Iterative imputation prevalent method for missing data imputation, where each feature is imputed iteratively by treating as target variable estimated from all other features. However, iterative imputation method suffers from two principal limitations: ① it imposes single parametric model form to impute all features, neglecting the potential for optimal models to vary among features, which risks model misspecification; and @ it assumes every feature contains missing values, overlooking the potential presence of non-missing features, termed as oracle features, which are informative for imputation. To address these limitations, we propose kernel point imputation (KPI), a bi-level optimization framework for iterative missing data imputation. At the inner level, KPI adaptively learns the optimal model form for each feature within a reproducing kernel Hilbert space, addressing limitation O. At the outer level, KPI utilizes oracle features as supervisory signals to iteratively refine the imputations, addressing limitation e. Experiments demonstrate that KPI outperforms competitive imputation methods. Code is available at https: httpp::///gtttit github...... com/FMLYD/kpi. git.
Introduction
Missing data are ubiquitous in real-world data collection and analytics [25, 46, 39]. For example, in manufacturing, temperature sensors may fail due overheating electrical disruptions, compromis- ing data integrity and impeding analytical workflows [1]. Similarly, equipment-monitrrine systems can experience connectivity loss in electrical sensors, impeding fault detection and introducing security risks [36, 35]. These issues highlight the importance of missing data imputation (MDI) techniques, which aim recover missing data using observed data, thereby enhancing the integrity of collected datasets [40, 38] and the reliability of subsequent data-driven applications [3, 2, 33, 43].
Existing MDI methods can be broadly categorized as discriminative or generative [7]. On the one hand, discriminative methods, such statistical imputation (e.g., mean and median imputation [45]) and iterative imputation (which iteratively estimates missing values using univariate models [27, 30]). have been well developed. On the other hand, generative methods have recently attracted attention
"This work was done in internship at Xiaohongshu Inc. Both authors have equal contribution.
i Corresponning author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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for their capacity to model complex data structures [1]. However, they often encounter training challenges [19, 26] or rely on strong data assumptions [42. 28]. Empirically. generative methods are frequently outperformed by discriminative methods [46, 24]. Therefore, discriminative methods
remain the preferred choice for MDI practice [38].
Among discriminative methods, iterative imputation is widespread for its plain implementation and strong performance. This approach specifies univariate model for each feature conditioned on the rest and iteratively updates missing values until convergence [27]. Despite its popularity,
standard iterative imputation faces two significant limitations. 1 it imposes a single parametric model form to impute all features, which risks model misspecification, as features often exhibit heterogeneous dependencies that cannot be adequately captured by a fixed-form parametric model [7]. @ it assumes every feature contains missing values, which neglects the potential of oracle features, i.e, non-missing features, to serve as high-quality supervisory signals for imputation.
To counteract these two limitations, we reformulate iterative imputation as a bi-level optimization problem. To address limitation O, the inner-level optimization adaptively selects functional forms from a reproducing kernel Hilbert spaces (RKHS) for each feature, reducing model misspecification. To address limitation @, the outer-level optimization aligns the imputed values with the oracle features, leveraging them as direct supervisory signals. Subsequently, we propose kernel point imputation (KPI), which expresses the optimal model as a linear combination of kernel functions, enabling efficient solution via stochastic gradient descent. Furthermore, we design an adaptive kernel ensemble strategy to dynamically combine kernels, thereby enhancing model expressiveness and alleviating hyperparameter selection challenge amidst incomplete data.
Contributions. The key contributions of this study are summarized as follows:
We identify two critical limitations in current iterative imputation methods: it imposes a single parametric model form to impute all features, which risks model misspecification, and @ it assumes every feature contains missing values, which overlooks the utility of oracle features.
We introduce KPI, bi-level MDI framework that addresses the identified limitations while remaining compatible with gradient-based optimizers. Additionally, we develop a kernel ensemble strategy to address the difficulty of hyperparameter selection amidst incomplete data.
We conduct extensive experiments to demonstrate the superiority of KPI over competitive baseline methods. Our results also confirm that both the model form optimization and the utilization of oracle features can effectively enhance imputation performance.
2 Preliminaries
This study focuses on the MDI problem an end goal. We do not treat imputation as preprocessing step for downstream tasks [7], such as time-series forecasting [33] or lsbudo-labeling [12], as these settings typically require joint optimization with task-specific objectives [17].
The MDI problem is specified by the following core components: Ideal matrix X (id) E NxD, the
non-missing (ideal) data matrix with N samples and D) features; @ Missingness matrix M E RND,
a binary matrix where Mn,d = 1 indicates missingness and Mn,d = O otherwise; Observed matrix (obs) E RRND: the data matrix containing missing values, expressed as X (obs) (id) (
- (1-M)- + nan O M; O Imputation matrix X (imp) E NxD; the matrix that replaces the missing values in X (obs) with imputed values. On this basis, the task of MDI is to construct an imputation matrix X (imp) based on X (obs) that closely approximates (id) Different imputation methods differ in how they map X (obs)
X (imp).
prevalent family of methods is iterative imputation, which iteratively treats each feature as a target variable and estimate its missing values from the remaining features. In the training stage, for arbitrary
1 & d < D, define the target feature Y (obs)) x .d
(obb)) The method fits an imputation model fo that learns the dependency between the target feature and the other features by solving
(obs) fx (obs)
min
(1)
where X (bbs) denotes X (ohs) with the d-th column removed. This procedure is repeated for d
1.... D, yielding D univariate imputation models. In the inference stage, imputation is performed

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K
5.
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7.5
1000
82
%
X4
X2
(a) The implication of model misspecification.
(b) The implication of overlooking oracle features.
Figure 1: Case study illustrating the limitations of iterative imputation. In panel (a), circular and cross markers indicate observed and missing values, respectively, while lines represent imputation model outputs. In panel (b), "v" denotes whether sample be used for training the imputation model for Y; dark areas indicate missing indices in Y at missing ratios of 50% (left) and 75% (right).
iteratively: for each target feature, its missing values are estimated using the corresponding univariate model, conditioning on the current values of the remaining features (including values imputed in earlier steps). The resulting imputed features form the imputation matrix X (imp)
3 Methodology
3.1 Motivation
In this section, we discuss two limitations inherent to standard iterative imputation. It imposes single parametric model form to impute all features, such as a linear model [27] or decision tree [30]. This rigidity risks model misspecification, as the optimal functional form often varies across features due to heterogeneous dependencies with other features [7]. For example, in manufacturing process, temperature may depend linearly on pressure, while vibration could depend on pressure nonlinearly. A fixed parametric form cannot accommodate such heterogeneous dependencies, resulting in suboptimal imputation performance. values. This assumption ignores the potential of oracle features-featuree that fully nearly non- missing-to to provide high-quality supervision for the imputation process. In practice, oracle features ubiquitous, such demographie variables in healthcare records reliable sensor readings in industrial systems. By neglecting these features, current iterative imputation suffers from limited supervisory signals, particularly regimes with high missing ratios.
It assumes every feature contains missing
Case study. To substantiate the claims above, a case study is conducted. On limitation 1, Fig. (a) demonstrates that a fixed model form leads to model misspecification. In the left panel, linear model accurately fits the linear feature well but underfits the others; in the right panel, nonlinear model fits the sine feature well but overfits the others. Therefore, enforcing single parametric model form risks model misspecification that limits imputation performance. On limitation @ Fig. 1 (b) illustrates that overlooking oracle features reduces the sample size usable for training fo- To impute the target feature Y, iterative imputation fits fa using X1,... X, as inputs, and trains only on samples where Y observed. Under high missingness, only handful of samples remain (two in the right panel), which is inadequate for learning reliable estimation model. In contrast, XI.... X4 are non-missing oracle features, yet the method does not leverage them provide additional supervision, thereby missing an opportunity to improve imputation accuracy.
These limitations underscore the need for improved iterative imputation approach. In particular, there are three key questions that warrant investigation: (1) How to select the optimal model forms for each feature reduce model misspecification? (2) How incorporate oracle features for training imputation models? (3) Can these improvements enhance imputation performance practice?
3.2
A bi-level optimization framework for iterative imputation
To address the two limitations above, propose novel bi-level optimization framework for iterative imputation. The framework learns feature-specific model forms within a reproducing kernel Hilbert space (RKHS) and integrates oracle features explicit supervisory signals.

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To address limitation O, we replace the fixed parametric estimator (fee in (1)) with nonparametric function optimized in an RKHS H. Taking the d-th feature the target feature, we reformulate the imputation task as regularized functional optimization problem:
( - f(x ((x(( || + + with Y =X x((bxx)
arg min SEN.
where f the optimal estimator for imputing the d-th feature. By optimizing over H, the estimator adapt its form to each feature, reducing the risk of misspecification.
To address limitation 0, incorporate oracle features supervisory signals for imputation.
Suppose (imp) X (obs) + X (miss), where X (miss) contains the imputed values be learned on missing entries (M- = 1) and is fixed to nan elsewhere. For an oracle feature Yooos)) with
its corresponding optimal estimator f", we refine the imputations of the remaining features by updating only their missing entries while keeping observed values fixed:
y((bb)) - ff" (x( (1mp)
min (imins)
(3)
This formulation is motivated by the following perspective: if X (imp) is correctly imputed, the optimal estimator f should produce estimates consistent with (abs), Conversely, large estimation errors indicate that the current X (imp) violates the dependency encoded by f*. This process exploits oracle features as effective supervisory signals to update the other features.
Suppose X = -x (lmp) and (abs)
x (obs) combining (2) and (3) immediately yields:
(obs) - f(x) + 1
min
min FEH
X (inis))
which is performed for d = D, treating each feature (including oracle features) in turn as the target feature (obs), This bi-level optimization addresses both limitations of iterative methods.
3.3 Kernel function, universal property and learning objective
To solve the inner loop in (4), approximate the optimum estimator f* via Gaussian kernels. We start by clarifying key properties of kernel functions Definition 3.1 and 3.2.
Definition 3.1 (Kernel function). Let X' be a non-empty set. A function K : X x X' is kernel function if there exists a Hilbert space H and feature map tb X H such that Vxr, x' E X. K K(x, = (z(x), (xx')H
Definition 3.2 (Universal kernel). For A' a compact Hausdorff space, a universal kernel ensures that any continuous function e X' + R can be approximated arbitrarily well within the RKHS H. Specifically, for any 0, there exists f E such that: sup,e. (f(x) e(x) S
The Gaussian kernel is defined by K(x, x exp(- x'12 2) and is universal in the sense of Definition 3.2 [29]. implies that the RKHS of Gaussian kernel, defined as H = span [ (1, xx) E X), admits uniform approximation of any continuous function.
Lemma 3.3 (Representer theorem). Suppose h(llf)) RR - IR is a non-decreasing function. The minimizer of an empirical risk functional regularized by h(I.fl)) admits the form: f*(()=
Σx_ a x24) where a (aI...., and K is the associated kernel function.
Lemma 3.4. Let Y Y E RBB1 be the target feature and X X be the corresponding input features; Suppose f is the optimal model minimizing the empirical risk in the inner optimization of (4), its output on X is given by f (x*) = Kx OG, where a = (K + XI) -Y;; Kx-x- is the kernel matrix computed with X and X
Since H may be infinite- dimeni--iimesional,, directly identifying f" E H is intractable. Lemma 3.3 yields finite-dimensionall characterization f³, and Lemma 3.4 further implies that, given two batches of target features (Ys, Y') and input features (X*, X¹), the outputs of f* on X admit the closed form:
f (x')= Kx (xxx + +AI 'Y".
(5)

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1
3.2
2
3.2
2
Calculate
0.1
2
6
2
6
4
0.4
3
X Y
1.8
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1.8
6
4.2
0.2
0.8
3
2
Sample
0.8
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2.2
0.2
(1mg)
X', Y
0
0
0
Gradient masking
Fxr,
Fxt, ?
Update imputation
0
0
Vx
3P
Ext,
Ext,
xt. 2
Vxt.
2
Ry. Pxt,
Ryt P
Vxy, 2
Forward pass
pass
features
Missing indices
Figure 2: Visualization of the workflow of KPI, where the dataset contains samples and 4 features. The sampling batch size set to 2. The last column is the oracle feature without missing values.
Therefore, the output of f* is expressed combination of kernel evaluations, retaining feature-wise adaptation of model form while reducing (4) to a differentiable loss function:
Y - Xxx + AI) Y 2
(6)
The tuning of kernel hyperparameters (e.g.. the Gaussian bandwidth) is notably challenging amidst incomplete data. To alleviate this problem, we employ multiple kernels with distinct parameters and learn ensemble them adaptively. Suppose K K'K K., K are kernel matrices with different
hyperparameters, E RK is a learnable simplex vector; we form the ensemble kernel as K K A
KK Ag. Putting together, the final objective becomes:
p
Y
K X
(K
+ +XI YY)
3.4 Overall workflow
While the learning objective is well defined, how to use it for imputation remains unclear. this end, propose the kernel point imputation (KPI) method, which iteratively minimizes the objective (7) to refine imputation missing values. The core procedure is shown in Fig. and detailed as follows.
Initialization. Given the incomplete dataset (obs), initialize missing values using the mean of observed steps, obtaining an initial imputation matrix X kimp The imputed values are treated as learnable parameters, and their gradients tracked throughout training.
Forward Pass. Two batches are sampled from the imputation matrix. each iteration, a column randomly chosen the target feature (Y", Y' R. Bx1,, with the remaining columns input features (X', X E (D--)))) where B represents batch size, and differentiates different batches. The objective P is computed following (7).
Backward Pass. The gradients Pf with respect to X X and A are calculated using automatic differentiation. The imputed values in X and X well A then updated using gradient descent with an update rate
X X P
X - X P M
(8)
< VAP,
where only the missing values (with M 1) are updated, while the observed values (with M 0) remain unchanged during this process. Moreover, the gradient of the matrix inverse term is stopped for numerical stability. KPI iteratively executes the forward and backward passes sampling different batches until hitting the early-stopping criteria on the validation dataset. In this process, each feature iteratively treated the target feature while the remaining features are treated as the input features. oracle features-aa fully exploited as supervisory signals.
This ensures that all
5

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Table 1: Imputation performance in terms of MSE and MAE.
BT
CC
CBV
QB
WQW
Dalusets
Metrics
MSE
MAB
MSE
MAE
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAB
MSE
MAE
Mean
0.742
0,452
0,837
0.789
0.829
1.165
0.754
4.145
0.740
2.841
0.589
4.682
0.764
1.121
Mode
0.948
0.770
0.935
1,159
1,026
1.749
0.925
7,741
1,254
7,832
0.593
6,240
0.823
1.372
Median
0.706
0.884
0.820
1.165
0.713
4.356
0.698
3.029
0.500
5.066
0.756
1.123
MICE
0.580
0.127
0.745
0.474
0.856
1.021
0.733
4.539
0.417
1.312
0.536
3,415
0.824
0.971
Miss.F
0.560
0.241
0.732
0.650
0.764
0.994
0.593
3,277
0,526
1.497
0.436
3.202
0.686
0.898
Sinkhorn
0.835
0.466
0.906
0.796
0.898
1.225
0.848
4.945
0.827
3,233
0.775
6.114
0.857
1.170
TDM
0.730
0.487
0.819
0.769
0.799
1.113
0.726
3.965
0.722
2.792
0.570
4.756
0.752
1.098
CSDI-T
0.726
1.870
0.849
2.683
0.821
3.802
0.761
15,493
0.731
12.291
0.575
19.919
0.780
4.084
MissDifT
0.719
1.332
0.840
1.699
0.816
3,523
0.749
13.432
0.728
14.462
0.564
23.320
0.758
5.184
GAIN
0.730
0.396
0.777
0.688
0.729
0.942
0.572
3.318
0.448
1.413
0.476
4.669
0.754
1.095
MIRACLE
0.795
0.487
0.305
0.831
1.154
3.208
45.816
36.784
3.975
0.555
0.685
MIWAE
0.582
0.266
0.746
0.630
0.807
1.071
0.636
4.118
0.525
1.804
0.475
4.977
0.657
0.844
Remasker
0.439
0.131
0.767
0.750
0.528
0,522
0.599
3.584
0.447
1.268
0.401
2.811
0.546
0.636
NewImp
0.465
0.177
0.412
0.292
0.405
0.401
0.431
2.495
0.320
0.857
0.332
2.992
0.497
0.692
KPI(Our))
0.397
0.284
0.402
0.394
0.400
2.387
0.747
0.264
0.491
0.685
Note: Each entry represents the average results at four missing ratios: 0.1, 0.2, 0.3, and 0.4. The best and second-best results are bolded and underlined, respectively.
4 Experiments
4.1 Experimental setup
Datasets:
The empirical study is performed on public tabular datasets from [1], including Blood Transfusion(BT), Concrete Compression (CC), Connectionist Bench Vowel (CBV), Ionosphere (IS) Parkinsons (PS), Qsar Biodegradation (QB), and Wine Quality White (QWQ). We simulate missing data via a binary mask matrix, characterized by predefined missing ratio. The specific generation strategies of mask matrix in different missing scenarios are detailed in Appendix B.
The performance of KPI is compared against various imputation methods, including iterative imputers (MICE [27], Miss.F. F. [30], and generative models (GAIN [42], MIWAE [20], Miss. [25], CSDI-T [32], ReMasker [4], and New [1].. We also assess methods that do not conform to these categories, such as MIRACLE [9], Sinkhorn [24], and TDM [46].
Baselines:
Implementation details: To ensure convergence, we cap the number of iterations 500 and adopt an early stopping criterion based on validation performance, with a patience of 10 epochs. The Adam optimizer used for training [8]. Key hyperparameters, namely n and B, are determined by allocating 5% of the training data for validation and finetuning over (0.0001, 0.01] for ' and [64, 512] for B. Performance assessed using modified mean absolute error (MAE) and mean squared error (MSE), focusing on the imputed values at missing values, following [46, 7]. In addition, we report the distribution discrepancy (WASS), measured the Wasserstein distance [7]. The experiments are performed on platform with two Intel(R) Xeon(R) Platinum 8383C CPUs @ 2.70GHZ and a NVIDIA GeForce RTX 4090 GPU.
4.2 Overall performance
Tab. 1 presents the average imputation results of KPI and baseline methods under missing ratios Pmiss 0.1, 0.2, 0.3, and 0.4. Key observations are summarized as follows:
The iterative imputers exhibits promising performance in most cases. For instance, MICE outper- forms simple imputers by large margin over most datasets. MissForest employs random forest as the base model, excelling in handling tabular data, which further improves imputation quality.
The canonical generative imputers [32, 25), originally tailored for time-series data, often falling behind iterative methods. This can be attributed to the implicit maximization of imputation

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0.9
0.5
0.7
WASS
0.2
0.2
0.1
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.5
0.8
1.0
02
04
0.6
0.8
1.0
Ratio of oradle features
Ratio of oradle features
Ratio of oracle festures
Ralio of oracce festures
(a) The results on the CC datasct.
(b) The results on the CBV dataset.
0.5
0.7
MBBF
MAE
0.5
WASD
0.5
033
0.2
0,2
0.3
0.1
0.2
0.4
0.2
0.4
0.5
0.4
0.5
0.8
0.5
0.8
1.0
Rallo of oradi fealures
Railo oracle features
Relio of oracle features
Ralio onacin fastures
(c) The results on the IS dataset.
(d) The results on the QB dataset.
Figure 3:
The performance Miss. and KPI given varying ratios oracle features.
Table 2: Varying kernel number results.
Table Varying kernel function results.
CC
CC
AMSE
AWASS
AMAE
AMSE
AMAE
0.082
0.058
0.155
Lincar
0.099
0.051
0.203
D.070
14.6%%
0.055
5.29&
0.116
25.2%
Poly
0.065
34.3%
0.039
23.58.
0.091
55.2%
0.069
15.9%
0.046
20.7%-
0.108
30.39L
Taplacian
0.068
31.3%%
0.042
17.6%..
0.093
54.2%
7
0.065
20.7%8.
0.039
32.8%-L
0,091
41.398.
Gaussian
0.076
23.2%
0.035
31.4%%
0.082
$9.618.1
CBV
CBV
MSE
AMSE
WASS
AWASS
MAE
AMAE
Kemel
MSE
AMSE
WASS
AWASS
MAE
AMAE
0.095
0,233
Linear
0.089
0.087
0.219
0.110
14.1%
0.085
10.5%L
0,226
3.0%
Poly
0.087
2.2%
0.088
1.1% f
0,214
2,3%
0.098
23.4%
0.075
21.15-L
0,216
7.3%5.
Laplacian
0.083
6.7%
0.080
8.1%
0.211
3.7%
0.087
32.0%
0.066
30.3% L
0.205
12.098.
Gaussian
0.087
2.2%
0.066
24.19%
0.205
6.4%L.
BT
BT
MSR
AMSE
WASS
AWASS
MAE
AMAE
Kernel
MSE
AMSE
WASS
AWASS
MAE
AMAR
0.109
0.363
Tinear
4.8%8]
0.101
7.3%
0.352
3.0%
Poly
0.305
6.4%)
10.9%
0.342
4.7%
0.305
8.7%
0.096
11.95.
0.343
5.5%5.
Laplacian
0.316
3.1%
0.091
9.9%
0.346
3.6%L
0.302
9.69E.
0.089
18.35-L
0.338
6.9%8..
Gaussian
0.302
7.4%.L
0.089
11.9%.
0.338
5.8%
entropy in diffusion models, which negatively impacts accuracy [1]. By contrast, recent generative approaches such as NewImp and Remasker handle this issue and achieve strong results, obtaining the best results among baseline methods.
KPI improves the iterative imputers by adaptively selecting the optimal imputer for each feature and involving oracle features as supervisory signals. This strategy consistently improves performance, evidenced by KPI outperforming all baselines across all 7 datasets-ofiet by substantial margin, particularly on the CC and QB datasets-demonstrers strong practical effectiveness.
4.3 Impact of oracle features
In this section, we assess the impact of using oracle features as supervisory signal on imputation performance. Specifically, we simulate based complete datasets to generate varying ratios of oracle features and evaluate the imputation performance. Two models are considered: KPI and another canonical iterative imputer: Miss.F.
The results are presented in Fig. 3. As the ratio of oracle features increases, KPI consistently exhibits lower imputation error, showcasing the utility of oracle features. In contrast, Miss.F shows little improvement as oracle feature ratio increases. This difference arises because Miss.F only uses oracle features inputs, whereas KPI exploits them supervisory signals to refine the imputation results.

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0.5
0.
MAE
WIASS
MAE
0.5
0.005
0.001
0.005
0.01
0.02
0.0006
0.001
0.006
0.01
0.02
0.0006
0.001
0.005
0.01
0.02
0.0008
0.001
0.005
0.01
0.02
0
Varying learning rate results CC
(b) Varying learning results CBV
0.50
MAE
WASS
0.0
255
1024
255
1004
255
1024
255
B
B
(c) Varying batch size results on CC
(d) Varying batch size results on CBV
Figure 4: Varying learning rate and batch size results with missing ratios 0.1 and 0.2.
4.4 Impact of kernel strategy
In this section, we analyze the impact of kernel function and kernel amount (E) on imputation performance. The key observations are summarized as follows.
The multiple kernel ensembling mechanism has substantial impact. As shown in Tab. increasing E from to 7 consistently reduces MSE from 0.082 to 0.065, indicating a relative reduction of 20.7%. This gain is attributed to the increased flexibility in adaptively selecting kernel parameters, allowing KPI to better represent the optimal imputation model for each feature.
The performance of different kernel functions showcases the importance kernel universality. The linear kemel,, which is not universal and has limited RKHS capacity, yields the worst performance. The polynomial kernel, with larger RKHS, performs better. The Gaussian kernel exhibits the best overall performance. The superiority is attributed to its universality, i.e., the associated RKHS admits uniform approximation of any continuous function. Such extensive RKHS capacity enables KPI to optimize the imputation model for each feature, thereby enhancing imputation performance.
4.5 Parameter sensitivity analysis
In this section, we examine the influence of critical hyperparameters the performance of KPI in Fig. 4. Below are the key observations:
The update rate (1)) plays pivotal role in controlling the volume of updates to the imputation matrix each epoch. As 7) is reduced from 0.02 to approximately 0.01, both MAE and RMSE decrease, indicating that smaller 7 enhances update stability. However, further reduction of to 0.001 results increased MAE and RMSE, where the meaningful update direction becomes overshadowed by noise, preventing model convergence within the allocated epochs.
The batch size (B) affects the scale of the problem in calculating discrepancies, with sizes ranging from 64 to 1024 examined. There is weak yet consistent decrease in MAE and RMSE as the batch size increases to B = 512, enhancing the reliability of estimations. Increasing the batch size beyond this point yields diminishing returns and may lead to unnecessary computational overhead.
5 Related works
The pervasive presence of missing data undermines the integrity of collected datasets and the reliability of data-driven applications, underscoring the necessity for effective missing data imputation (MDI). To achieve accurate MDI, existing approaches can broadly categorized into two paradigms: discriminative and generative, each with distinct advantages and limitations [7, 22).
The iterative method [30, 27, 44, 16] one of the most popular methods discriminative imputation, initiated from imputation by chained equations (ICE) [27], which employs specific models to estimate

---

missing values for each feature based on the remaining observable features. On the basis of ICE, line of work advocates for employing modern parametric models, such as neural networks [20, 9], Bayesian models [27] and random forest [30], which enhances the capacity of imputation models and thereby accommodating complex missing patterns. In a different line of work, various training techniques are investigated within the paradigm, such as multiple imputation [27], ensemble learning [30], and multitask learning [20], which enhance its utility in diverse contexts. While this paradigm offers enhanced flexibility and accuracy, it fails to utilize the oracle features effectively and risks model misspecification, which can lead to suboptimal imputation results in noisy environments. Our research advances this methodology by handling the two limitations.
Apart from the iterative methods, there are other notable approaches in the discriminative paradigm. The simple direct paradigm employs elementary statistical measures like mean, median, and mode to replace missing values, offering quick and straightforward solutions. However, this approach lacks the capacity to accommodate complex dependencies [18, 21J, often producing trivial and inadequate imputation results that fail to meet expectations in practice. Another notable approach is matrix factorization, which decomposes the data matrix into two low-rank matrices, capturing the latent structure of the data for imputation [10, 6]. This method particularly effective in collaborative filtering and recommendation systems [14, 37]. Recent advances explore novel methodology based on distribution discrepancy minimization[46, 24]. This approach builds on the assumption that, under the independent and identically distributed (i.id.) condition, any two data batches should share the same underlying distribution, thereby exhibiting minimal discrepancy. Subsequent studies have extended this idea by refining discrepancy measures accommodate different data characteristics such as neighboring effects [39, 35], noisy observations [36], and temporal dependencies [38].
The generative paradigm restates imputation as conditional generation problem, using advanced generative models, such as generative adversarial networks [42, 31, 15] and diffusions [32, 41, 1J, to approximate data distributions and perform imputation. This strategy incorporates the strengths of generative models, capturing and utilizing complex dependencies, which potentially enhances the imputation quality when ample data is available. However, it also bears the defects with generative models, such as the instability associated with adversarial training and the operational complexity of diffusions [19, 26], hampering their use in practice.
6 Conclusion
Iterative imputation methods are widely used for handling missing data, yet existing approaches often limited by model misspecification and underuse of oracle features. To overcome these limitations, we introduce KPI, a li-level optimization framework which optimizes model form within RKHS for each feature, reducing model misspecification, and exploits oracle features as effective supervision. Extensive experiments on real-world datasets demonstrate that KPI achieves superior imputation performance and effectively leverages oracle features.
Limitations & future work. In this work, we do not accommodate potential noise in datasets, which is prevalent challenge in industrial settings [6, 5]. Future research could incorporate robust optimization techniques and truncate outliers in the kernel matrix which has potential to improve noise robustness. Additionally, this work mitigates the difficulty of concise kernel parameter selection via adaptive ensembling, which is an heuristic approach. Subsequent work may explore meta-learning strategies with theoretical guarantees for accurate kernel parameter selection.
Acknowledgments
Z. Lin was supported by the NSF China (No. 62276004) and the State Key Laboratory of General Artificial Intelligence. H. Li was supported by National Natural Science Foundation of China (623B2002).
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12

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A Theoretical justification
In this section, we delve into the theoretical underpinnings of the KPI framework. By exploiting the universality, injectivity, and reproducing properties of kernel functions, we derive formal guarantees for KPI and demonstrate how these properties translate into effective imputation.
Lemma A.1 (Representer theorem). Suppose h(llfl) R. + I. is a non-decreasing function. The minimizer of an empirical risk functional regularized by h(ll.lll admits the form: f*(+) E a; (;,(;; where a -(a1...,,,,)) and K is the associated kernel kernel function.
Proof. The proof can be found in Theorem 6.11 of Mohri et al. [23].
Theorem A.2 (Lemma 3.4 in the main text). Let Y Y E Bx1 be the target feature and X*, X be the corresponding input features; Suppose f is the optimal model minimizing the empirical risk in the inner optimization of (4), its output on X is given by (*'X)) = Kx.x a, where xx is the kernel matrix computed with X and X (K + XI)
=
Proof. This is standard theoretical result in kernel ridge regression, and we detail its proof here with contextualization. Consider the samples X and Y where Y is the observed target, and X comprises the input features. The empirical risk minimization objective with &g regularization to select the optimal functional form is
min YY + Alffl
(9)
FEH
which corresponds precisely to the inner loop of (4). According to Lemma A.I, when is an identity function (in (9)) and H is RKHS associated with kernel K, the minimizer f* must admit the explicit form
-x)== a((Iz,)))
(10)
for some coefficients a1....,
Substituting this form into the empirical risk (9), the optimization problem becomes
min CERB
Y
Xx + Aa Kx
(11)
where Kx-x- is the B x B Gram matrix, with (i,j)-th entry K K(a), Y is the length-B target vector, and the vector of coefficients.
Expanding the loss function matrix notation yields
xxxxaa) (Y" Kx T xx
(12)
Due to symmetry of Xx this simplifies
Y Y - 2Y Kx-x-a a (K + XKx-x-
(13)
According to the first-order condition, setting the derivative with respect to a to zero and solving for a gives
2K x Y 2K*xxx XKxxx= 0,
(14)
which is equivalent to:
K x (Kx-x AI)a Y
(15)
Assuming Kx-x invertible, we have:
(Kx-x + AIJa Y
(16)
which immediately follows from multiplying both sides by K K Solving for a gives:
=KXx -
(17)
Substituting (17) into (10) leads to
B
[*(x) = Σ og (()() = (Kx) + +
(18)
13

---

where K(x) is the x B vector ( (x, For a (possibly distinct) batch of inputs X evaluating f at each zjj gives
a K x4) [f) ((x) (2), ((,,, xB]] (Kx-x -y, (x11) Σ
(2(bh) = Σ ar (02, xt)) - [K(a), Kuh,, K (Kx.x x1)
(x(b) = Σ (2 xt) = [K(r}. K() (2) (()b. (Kx.x- + +X
(19)
which may be stacked to give the vector-valued expression
(X') = x (xx.x + AI) -I) YY.
(20)
where X is the matrix with entries K X K(a), 2z). The proof completed.
L
Definition A.3 (Kernel Functions). X, RD be vectors the input feature space. A kernel function functtinn K RD x RD + IR is a symmetric, positive semi-definite function that quantifies the similarity between x and x'. Commonly used kernel functions include:
1. Linear Kernel: Kinner (x, x') x Tx', which computes the inner product between two vectors and corresponds to the case where no explicit feature transformation is applied.
Kpoly(x, x') = (x xxx'= c) where c 2 0 is a constant coefficient trading off the influence of higher-order versus lower-order terms, and d is the degree of the polynomial. It enables learning non-linear dependencies by implicitly mapping the input features into a higher-dimensional polynomial feature space.
Polynomial Kernel:
Gaussian Kernel: Kgauss (x, x') = exp
Xx 1xxx'1²" where ||x - x'I22 denotes the squared 20t
Euclidean distance between x and x', and > is scale parameter controlling the width of the kernel. The Gaussian kernel is widely used due to its ability to model localized and highly non-linear interactions.
B
Implementation details
B.1 Dataset description and process strategy
In this paper, we use UCI datasets for model validation. To simulate missing data, we first construct a binary mask matrix M. The observed data matrix, denoted as X(obe), is derived by element-wise application of the complement mask M the non-missing data matrix (id), On the generation of M, we consider three canonical missing data mechanisms:
Missing Completely at Random (MCAR): The probability of entry-wise missingness is indepen- dent of both observed and unobserved data. To simulate MCAR, each entry of M is independently set to 1 (missing) with probability Pmiss and to (observed) with probability 1 - Pmiss
variable depends only on values of observed variables [34, 11]. To generate MAR scenarios, we randomly select subset of features to be always observed. The missingness the remaining features is simulated using a logistic regression model, where the observed features act inputs. The model parameters are randomly initialized, and the bias calibrated to yield the desired missing ratio.
Missing at Random (MAR): The missingness
Missing Not at Random (MNAR): The probability that value missing depends on the values themselves [37, 13]. For MNAR simulation, we adopt the procedure in [1, 7]: the logistic model used for MAR repurposed, but its inputs are themselves masked by an independent MCAR mechanism, making the missingness dependent both observed and unobserved features.
14

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HEP- gaussion Briurr
poty
Teae (re)
2.
To (maa)
20
-B-
geusskan
Tie (ns)
4.
-0h gov12 on
Incor
Irev
poly
30
poyy
52
256
512
1024
32
256
512
1024
16
22
64
125
255
52
125
290
Batch
Batdh G20
amound (D)
amount
(a) Computation time with different batch sizes.
Computation time with different feature amounts.
Figure 5: Running time of the forward pass (left panels) and backward pass (right panels) given varying settings. Different colors indicate different kernel functions. The colored lines and the shadowed areas indicate the mean values and the 99.9% confidence intervals.
B.2 Training protocols
To ensure reliable convergence, set maximum of 500 training iterations and adopt early stopping based on validation performance, using a patience parameter of 10 epochs. Optimization throughout the Gaussian kernel. conducted using the Adam optimizer [8]. The kernel function is specified
The main hyperparameters, specifically the update rate 1), batch size B, kernel number E. and variance are determined by allocating 5% of the training data as validation set and tuning over the intervals E [0.0001, 0.01]. B E (64, 512]. E E 7] and a E [0.01, 10). All experiments are conducted on hardware platform comprising two Intel(R) Xcon(R) Platinum 8383C CPUs (2.70GHz) and an NVIDIA GeForce RTX 4090 GPU.
B.3 Evaluation metrics
The imputed data matrix (imp) is evaluated to assess imputation quality. Following the protocol in [46], we primarily employ the modified mean absolute error (MAF) and root mean squared error (RMSE) for evaluation:
MAE
(imp) And
x.
(abs)
Madd
(21)
Σx
D 2d=1
Mad n=1 d=l
ΣΣ
(imp) (obs))
||2 and
RMSE
and
Mind
(22)
En=1 dd=1 Mid h=1==1
where Mind E M indicates positions of imputed (originally missing) values with Mnd = 1 mnd- and x and
(obs) X (obs) is the ground-truth value from the non-missing data. As only the originally missing values are imputed, we restrict the calculation of error metrics to the indices where m,, 1.
In addition the point-wise
metrics above, we also consider the squared Wasserstein distance (abbreviated as WÅSS) [1], which quantifies the discrepancy between the distributions of the imputed values and the corresponding ground-truth values at the missing positions (M 1).
C Additional experimental results
C.1 An empirical analysis on complexity
In this section, examine the practical computational complexity of KPI, typically the computational cost per iteration, which includes both the forward and backward passes. We conduct experiments using Intel@ Xeon@ Gold 6140 CPUs and Nvidia RTX 4090 GPUs, with each experiment repeated 100 times to ensure reliability.
The results are presented in Fig. 5. The running time per iteration remains limited (within 4 ms) across a diverse range of hyperparameters, demonstrating the fcasibility of KPI for real-world applications. Other key observations are summarized as follows:
To explore the impact of batch size (B), we vary within wide range from 32 to 1024 while keeping the feature amount (D) to 8. The running cost of the forward pass increased with the batch
15

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Table 4: Imputation performance comparison with missing ratio of 0.1.
BT
CC
CBV
QB
WQW
Dalusets
Metrics
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MICE
0.118
0.027
0.155
0.075
0.196
0.16
0.175
0.406
0.097
0.133
0.122
0.271
0.192
0.151
Miss.F
0.123
0.037
0.172
0.111
0.185
0.149
0.141
0.295
0.128
0.163
0.102
0.284
0.164
0.129
Sinkhorn
0.840
0.434
0.903
0.614
0.896
0.837
0.850
2,047
0.841
1.513
0.784
2.622
0.856
0.755
TDM
0.724
0.415
0.815
0.545
0.787
0.690
0.720
1.592
0.731
1,295
0.565
1.977
0.745
0.650
CSDI-T
0.727
1,914
0.850
2,680
0.815
3.753
0.766
16.714
0.743
12,939
0.578
20:407
0.775
4.022
MissDiil
0.718
1.446
0.847
1.803
0.812
4.101
0.750
13.640
0.744
16.209
0.566
25.062
0.755
6.037
GAIN
0.739
0.355
0.759
0.479
0.690
0.541
0.532
1.137
0.399
0.460
0.409
1.192
0.736
0.621
MIRACLE
0,528
0.174
0.382
0.161
0.778
0.682
3.723
26.666
3.777
18.544
0.461
1.103
0.485
0.364
MIWAE
0.539
0.226
0.698
0.436
0.782
0.668
0.603
1.638
0.526
0.861
0.450
2.044
0.626
0.507
Remasker
0.365
0.099
1.041
0.830
0.448
0.249
0.715
1.775
0.500
0.739
0.489
1.737
0.503
0.364
NewImp
0,383
0.273
0.110
0.231
0.423
1.013
0.281
0.305
1.067
1.045
KPI(Ous))
0.084
0.022
0.023
0.01
0.051
0,016
0.093
0.217
0.071
0.066
0.05
0.219
0.082
0.058
Table 5: Imputation performance comparison with missing ratio of 0.2.
BT
CC
CBV
1S
PK
QB
WQW
Datasets
Metries
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MICE
0.145
0.027
0.171
0.097
0.208
0.218
0.186
0.915
0.102
0.243
0.13
0.589
0.2
0.211
Miss.F'
0.141
0.058
0.173
0.131
0.187
0.206
0.144
0.589
0.133
0,316
0.106
0.579
0.168
0.181
Sinkhorn
0.834
0.428
0.907
0.711
0.902
1.079
0.842
3,908
0.819
2,572
0,773
5.036
0.854
1.030
TDM
0.725
0.431
0.812
0.659
0.800
0.939
0.720
3.097
0.710
2.167
0.567
3,855
0.750
0.927
CSDL-T
0.724
1,808
0.847
2.674
0.823
3.760
0.759
15.642
0.724
12.409
19.999
0.777
4.057
MissDDa
0.714
1.282
0.835
1.707
0.818
3.658
0.746
13:473
0.718
14.872
0.562
23.777
0.757
5.526
GAIN
0.727
0,350
0.759
0.585
0.701
0.739
0.526
2.231
0.409
0,830
0.407
2.292
0.724
0.853
MIRACLE
0.637
0,271
0.443
0.234
0.878
1.102
3.361
43.583
3.612
31.612
0.487
2.558
0.533
0.556
MIWAE
0.569
0.730
0.535
0.801
0.620
3.198
1.398
4.102
0.653
Remusker
0.403
0.108
0.412
1.223
0.488
0.392
0.613
2.959
0.450
1.077
0.397
2.482
0.523
0.531
Newlmp
0.441
0.141
0,360
0.201
0.310
0.221
0.411
1.937
0.283
0.592
0.329
2.273
0.468
0.265
KPI(O
0.095
0.032
0.077
0.051
0.085
0.064
0.098
0.436
0.075
0.128
0.062
0.322
0.103
0.11
size, as expected. This attributed the larger matrix inversion in (7), which cannot be efficiently accelerated by GPUs. In contrast, the backward pass cost was weakly correlated with B, since gradient computations after constructing the computation graph can be parallelized.
To investigate the impact of feature amount (D), we maintain constant batch size of 64. A weak correlation between D and the running time is observed. This is because varying D primarily affects the complexity of each kernel matrix computation, which can be effectively mitigated by GPU accelcration. This highlights a practical advantage of KPI: its efficiency in handling datasets with large number of features.
We observe that the type of kernel function also affects the running cost. The gaussian kernel exhibits the largest running time compared to other kernel functions, in terms of both forward pass and the backward pass.
C.2 Additional overall performance results given different missing ratios
Tab. 4-7 detail the imputation performance of KPI and baselines, with results for different missing ratios: 0.1, 0.2, 0.3, and 0.4 listed separately. The results demonstrate that KPI consistently outper- forms the baselines in all settings, achieving superior performance in terms of both MAE and WASS. This consistent superiority across varying missing ratios underscores the effectiveness and robustness of KPI for missing data imputation.
16

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Table 6: Imputation performance comparison with missing ratio of 0.3.
PK
QB
Dalusets Metrics
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
mice
0,156
0,043
0.195
0.133
0.219
0.289
0.186
1.393
0.107
0,387
0.134
1.044
0.21
0.28
missforest
0.14
0.064
0.189
0.182
0.2
0.292
0.154
1,049
0.131
0,423
0.113
1,085
0.176
0.262
0.475
0.853
1,368
6.014
0.774
0.859
1.313
idm
0.733
0.506
0.825
0.834
0.809
1.260
0.730
4.796
0.723
3.337
0.571
5.629
0.754
1.240
CSDI-T
0,717
1.905
0,851
2.684
0.826
3,816
0,761
14.942
0.729
12.044
0.574
19.732
0.782
4.093
MissDiil
0,718
1.317
0.842
1.656
0,822
3.313
0.751
13.341
0.725
13,806
0.563
22.714
0.759
4.894
gain
0.742
0.413
0.780
0.729
0.736
1.041
3.702
0.460
1.656
0.434
3.637
0.730
1.140
miracle
0
0.850
0.535
0.371
0.841
1.302
3.036
54.592
3.432
43.764
0.542
4.814
0.582
0.792
miwae
0.593
0.273
0,769
0.692
0.818
1,210
0.650
4.974
0.527
2,113
0.480
5.810
0.667
0.955
remasker
0.459
0.134
0,552
0.371
0.541
0.586
0.534
3,949
0.415
1.365
0.349
2,928
0.557
0.724
NewImp
0.341
0.443
3.066
0.350
3.666
0.558
0.379
KPI(Ous))
0.104
0.028
0.116
0.088
0.122
0.122
0.102
0.712
0.083
0.217
0.069
0.562
0.148
0.215
Table 7: Imputation performance comparison with missing ratio of 0.4.
CC
CRV
15
PK
QB
WQW
Datasets Metrics
WASS
MAE
MAE
MICE
0.162
0.03
0.223
06168
0.233
0.354
0.186
1.824
0.111
0.549
0.15
1.491
0.222
0.329
MISS.F
0,156
0,083
0.198
0.226
0.192
0.346
0.154
1.346
0.133
0.595
0.114
1,254
0.177
0.326
Sinkhomn
0.837
0.530
0.904
1.008
0.889
1.616
0.847
7,810
0.821
4.948
0.768
9.508
0.860
1.582
TDM
0.737
0.596
0.824
1.037
0.802
1.561
6.376
0.724
4.367
0.578
7.565
0.758
1.574
CSDL-T
0.735
1.851
0.847
2.694
0.817
3.877
0.757
14.671
0.727
11.771
0.577
19.537
0.786
4.164
MissDiff
0,726
1.284
0,837
1.628
0.812
3.019
0.750
13:272
0.723
12,960
0.566
21.728
0.761
4.278
GAIN
0.712
0.464
0.812
0.960
0.791
1.448
0.665
6.203
0.522
2.708
0.653
11.553
0.826
1.767
Miracle
1.401
0.602
0.483
1.529
58.421
3.250
53.215
0.595
7.425
0.620
1.027
MIWAE
0.629
0,344
0,786
0.856
0.828
1,502
0.670
6.663
0.535
2,842
0.504
7.952
0.683
1.185
Remasker
0,528
0,182
1.022
1.541
0.636
0.860
0.534
5.653
0.424
1.891
0.368
4.096
0.601
0.926
NewImp
0.563
0.301
0.553
0.520
0.542
0.743
0.451
4.035
0.351
1.563
0.378
4.989
1.022
1.542
KPT(Ours)
0.039
0.134
0.191
1.022
0.084
1.028
0.159
0.303
C.3
Additional overall performance results given different missing mechanisms
Tab. 8 and 9 provide a detailed evaluation of the imputation performance of KPI and various baselines under MAR and MNAR mechanisms, respectively. These missing mechanisms challenging compared to the MCAR setting reported in Tab. 1, but they are also more representative of real-world scenarios.
complex and
The results demonstrate that KPI consistently outperforms the baselines all settings, achieving superior performance in terms of both metrics. This consistent superiority across different missing mechanisms highlights the effectiveness and robustness of KPI for missing data imputation, making it reliable choice for diverse real-world applications.
17

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Table 8: Imputation performance comparison under MAR missing mechanism.
BT
CC
CBV
IS
PK
QB
WOW
Datasets
Metrics
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MICE
0.481
0.109
0.626
0.349
0.839
0.652
0.677
1,113
0.492
0.946
0.604
2.42
0.824
0.775
MISS.F
0.718
0.727
0.632
0.783
0.572
0.58
1.261
0.797
1.517
0.58
2.671
0.709
CSDI-T
1.094
5,465
0.894
3.212
0.826
4.286
0.707
15.194
1.262
19.116
0.782
23.176
0.815
4.919
MissDiff
1.019
2,835
0.888
2,189
0.852
6.008
0.704
13.233
1.219
22,773
0,762
34.125
0.805
6.919
1,082
1.187
0.782
0.570
0.700
0.503
0.456
0.609
0.709
1.383
0.552
1.716
0.729
0.672
MIRACLE
0.699
0.356
0.710
3.837
23.842
0.494
MIWAE
0.747
0.784
0.735
0.517
0.788
0.617
0.474
0.811
0.758
1.674
0.660
2.892
0.629
0.575
Remasker
0.598
0.689
1.023
0.854
0.467
0.235
0.691
1.149
0.668
1.062
0.557
1.563
0.489
0.397
Sinkhorn
1.106
1.319
0.958
0.718
0.923
0.804
0.829
1,350
1.257
3.392
0.904
3.032
0.905
0.912
TDM
1.015
0.855
3.311
2.739
NewImp
0.401
0.171
0.232
0.111
0.221
0.070
0.331
0.504
0.462
0.745
0.560
3,330
0.372
0.293
multikip
0.367
0.107
0.199
0.121
0.225
0.115
0.345
0.864
0.457
0.68
0.339
1.965
0.362
0.311
Table 9: Imputation performance comparison under MNAR missing mechanism.
BT
CC
CBV
IS
PK
QB
WQW
Datasets
Metrics
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MAE
WASS
MICE
0.696
0.411
0.66
0.384
0.832
0.734
0.708
1,831
0.499
0.98
0.589
2.377
0.807
0.724
MISS.F
0.731
0.697
0.748
0.587
1.944
1.513
2.291
0.667
CSDLT
0.885
3.105
0.885
2.923
0.838
3.922
0.759
16.833
1.016
14.173
0.683
20.330
0.795
4.275
GAIN
0,846
0.595
0.782
0.545
0.698
0.570
0.529
1.151
0.571
1.305
0.474
1.943
0.730
0.684
MIRACLE
0.655
0.319
0.371
0.163
0.842
0.802
3.725
27.093
4.196
25.052
0.576
2.249
0.518
0.437
0.658
0.735
0.559
1.535
1.400
Remasker
0.481
0.297
1.028
0.886
0.487
0.296
0.660
1.701
0.586
1.059
0.516
2.128
0.519
0.434
Sinkhorn
0.967
0.752
0.940
0.698
0.925
0.911
0.854
2.121
1.049
2,906
0.844
3.755
0.880
0.859
TDM
0.858
0.732
0.849
0.620
0.821
0.756
0.724
1.640
0.969
2.713
0.661
3.202
0.772
0.744
NewImp
0.645
0.585
0.562
0.442
2.328
7.161
multikip
0.573
0.257
0.271
0.22
0.357
0.352
0.391
1.344
0.423
0.769
0.286
1.967
0.458
0.563
18

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Reviews:
[{"review_id": "fwvS8vNv9T", "reviewer": "Reviewer_tbrx", "summary": "The paper presents an iterative imputation strategy based on bi-level optimization, proposed as an alternative to the existing MDI method. The approach consists of three steps: (1) randomly select a feature, (2) find a function in a Reproducing Kernel Hilbert Space (RKHS) that accurately predicts this feature using the others, and (3) update the imputation of other features to minimize the prediction risk associated with the selected feature. The method leverages oracle features—those with no missing data—and is empirically validated on a large set of datasets with artificially introduced missing values.", "questions": "You could address the points raised under 'Weaknesses' to improve your rating.\n\nOverall, the theoretical overstatements and the omission of the missing data mechanism are the main reasons for my current recommendation to bordeline reject the paper . However, these two issues are relatively easy to address, and given the novelty and practical interest of the method, my evaluation could be revised to \"accept\" if they are properly corrected.", "limitations": "yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### **Strengths**\n\n* The method is built upon a well-presented heuristic and offers an interesting alternative for practical applications.\n* The imputation literature is well-situated and contextualized.\n* The use of RKHS is clearly introduced, and the derivation of its solution is well explained.\n* A wide variety of datasets are used for empirical evaluation, which supports the general relevance of the method—although the study remains incomplete in some aspects (see weaknesses).\n\n### **Weaknesses**\n\n* Some of the theoretical results (in main) are incorrect or imprecise. In particular, key assumptions are missing or improperly stated—for instance, the Gaussian kernel is not convex in general, yet convexity is assumed without justification. There are overstatements in the theoretical justifications (e.g., convergence and rates are claimed without sufficient assumptions or guarantees). Moreover, the algorithm uses an approximate minimizer, not the true one, and the true minimization problem is non-convex.\n* The paper does not discuss the mechanism of missing data. The proposed method appears to be valid only under the MCAR assumption, which is the most favorable case. Experiments are conducted under this assumption, limiting the scope of comparison with methods that perform well under MAR  or MNAR .\n* The method seems to be more of a **completion** strategy than a true **imputation** method. It is unclear how the method could be used to impute missing values on *new* data points without retraining the model, unlike other methods such as MICE or MissForest. The experimental section should include a scenario with a held-out test dataset to evaluate generalization.\n* Notation is sometimes imprecise or not rigorous. For example, the function $f^\\star$ is applied to inputs that may contain Na values, which is not clearly defined or explained.\n\nTypos (no response needed): \n* $X^{imp}$ instead of $X^{(\\mathrm{imp})}$ (177)\n* Nan (Not a number )-> Na (not attribute)", "was_revised": true, "final_justification": "After rebuttal period the two main concerns are well addressed by the authors: rate: 3->5", "rebuttal": "Thank you very much for your constructive comments and appreciation of our heuristic and empirical studies. We are grateful that you have highlighted the key issues and expressed openness to rating adjustment. Below are our responses to the specific concerns and queries.\n\n---\n\n#### *[W1] Some of the theoretical results are incorrect or imprecise. In particular, key assumptions are missing or improperly stated—for instance, the Gaussian kernel is not convex in general. There are overstatements in the theoretical justifications (e.g., convergence and rates are claimed without sufficient assumptions or guarantees). The algorithm uses an approximate minimizer, not the true one, and the true minimization problem is non-convex.*\n\n**Response.** Thank you for your insightful comments on the theoretical analysis. We acknowledge the raised points and will revise our analysis accordingly.\n- On the convexity of kernel function. In the main paper, we assumed kernel convexity. For Gaussian kernel we demonstrated that it has a convex region given dimension $\\mathrm{D}=1$, but it is not convex in general cases with $\\mathrm{D}>1$. We acknowledge this imprecision and will now restrict our discussion to the linear kernel, which is convex.\n- On the overstatements like convergence assumptions. Given linear kernel, we immediately have the seperate convexity of the loss function to $X_t$ (since it is a standard quadratic function). Under these conditions, we can demonstrate convergence for the minimization process by updating $X_t$. \n> Considering the learning objective in (8) with linear kernel, define $g(\\mathbf{X}^t) = (\\mathbf{Y}^t - \\mathbf{X}^t \\mathbf{w})^2$ where $\\mathbf{w} = \\mathbf{X}^{s\\top}(\\mathbf{X}^s\\mathbf{X}^{s\\top} + \\lambda\\mathbf{I})^{-1}\\mathbf{Y}^s$. Suppose $\\mathbf{X}^t_k$ is the $k$-th gradient descent iteration to minimize $g(\\mathbf{X}^t)$, $\\mathbf{X}^t_*$ is the minimizer of $g(\\mathbf{X}^t)$. We make assumtions as follows:\n<!- > - (1) $g$ is $L$-smooth in $\\mathbf{X}^t$, i.e., for all $\\mathbf{X}, \\mathbf{Z}$ we have $\\|\\nabla g(\\mathbf{X}) - \\nabla g(\\mathbf{Z})\\|_F \\leq L \\|\\mathbf{X} - \\mathbf{Z}\\|_F$;->\n> - (1) $g$ is $L$-strongly convex in $\\mathbf{X}^t$, i.e., for all $\\mathbf{X}, \\mathbf{Z}$ we have $g(\\mathbf{Z}) \\geq g(\\mathbf{X}) + \\langle \\nabla g(\\mathbf{X}), \\mathbf{Z} - \\mathbf{X} \\rangle + \\frac{L}{2}\\|\\mathbf{Z} - \\mathbf{X}\\|_F^2$\n> - (2) the update step size is small, i.e., $\\eta \\leq \\frac{2}{L}$. \n> - (3) the initial value $\\mathbf{X}^t_0$ is proximal to $\\mathbf{X}^t_*$, i.e., $\\|\\mathbf{X}^t_0-\\mathbf{X}^t_*\\|_2^2 < \\epsilon^2$.\n\n> Then, for any $\\mathrm{K} \\geq 0$, we have$$g(\\mathbf{X}^t_\\mathrm{K})-g(\\mathbf{X}^t_*)\\leq \\frac{1}{2\\eta\\mathrm{K}}\\epsilon^2.$$\n\n\n> Proof. We first consider the iterative update using gradient descent:$$\\mathbf{X}^t_{k+1} = \\mathbf{X}^t_k - \\eta \\nabla_{\\mathbf{X}^t} g(\\mathbf{X}^t_k).$$ For a $L$-strongly convex function $g$, we derive the **core equation**:\\begin{equation}g(\\mathbf{X}^t_{k+1})\\leq g(\\mathbf{X}^t_k)+\\nabla_{\\mathbf{X}^{t}}^t g(\\mathbf{X}^t_k)(\\mathbf{X}^t_{k+1}-\\mathbf{X}^t_{k})+\\frac{L}{2}\\|\\mathbf{X}^t_{k+1}-\\mathbf{X}^t_{k}\\|^2\\\\ =g(\\mathbf{X}^t_k)-\\eta\\|\\nabla_{\\mathbf{X}^{t}}^tg(\\mathbf{X}^t_k)\\|^2+\\frac{L\\eta^2}{2}\\|\\nabla_{\\mathbf{X}^{t}}^tg(\\mathbf{X}^t_k)\\|^2\\\\ \\leq g(\\mathbf{X}^t_k)-\\frac{\\eta}{2}\\|\\nabla_{\\mathbf{X}^{t}}^tg(\\mathbf{X}^t_k)\\|^2 \\\\ \\leq g(\\mathbf{X}^t_*)+\\nabla_{\\mathbf{X}^{t}}^tg(\\mathbf{X}^t_k)(\\mathbf{X}^t_k-\\mathbf{X}^t_*)-\\frac{\\eta}{2}\\|\\nabla_{\\mathbf{X}^{t}}^tg(\\mathbf{X}^t_k)\\|^2 \\\\ = g(\\mathbf{X}^t_*)+\\frac{1}{2\\eta}\\|\\mathbf{X}^t_k-\\mathbf{X}^t_*\\|^2 - \\frac{1}{2\\eta}\\|\\mathbf{X}^t_k-\\mathbf{X}^t_*\\|^2 +\\nabla_{\\mathbf{X}^{t}}^tg(\\mathbf{X}^t_k)(\\mathbf{X}^t_k-\\mathbf{X}^t_*)-\\frac{\\eta}{2}\\|\\nabla_{\\mathbf{X}^{t}}^tg(\\mathbf{X}^t_k)\\|^2\\\\ = g(\\mathbf{X}^t_*)+\\frac{1}{2\\eta}\\|\\mathbf{X}^t_k-\\mathbf{X}^t_*\\|^2 - \\frac{1}{2\\eta}\\|\\mathbf{X}^t_k-\\mathbf{X}^t_*-\\eta\\nabla_{\\mathbf{X}^{t}}^tg(\\mathbf{X}^t_k)\\|^2\\\\ = g(\\mathbf{X}^t_*)+\\frac{1}{2\\eta}(\\|\\mathbf{X}^t_k-\\mathbf{X}^t_*\\|^2 - \\|\\mathbf{X}^t_{k+1}-\\mathbf{X}^t_*\\|^2),\\end{equation} where the first equation uses assumption 1; the second equation uses the update rules of the gradient descent; the fourth step holds as we set $\\eta \\leq \\frac{1}{L}$ which makes $- \\eta + \\frac{L \\eta^2}{2} \\leq -\\frac{\\eta}{2}$.\n\n> Summing up the inequality above with $k=0,1,...,\\mathrm{K}$, we have: \\begin{equation} \\sum_{k=1}^\\mathrm{K} g(\\mathbf{X}^t_{k}) - \\mathrm{K}g(\\mathbf{X}^t_*) \\leq \\frac{1}{2\\eta}(\\|\\mathbf{X}^t_0-\\mathbf{X}^t_*)\\|^2-\\|\\mathbf{X}^t_\\mathrm{K}-\\mathbf{X}^t_*\\|^2)\\leq \\frac{1}{2\\eta}\\|\\mathbf{X}^t_0-\\mathbf{X}^t_*\\|^2.\\end{equation} According to the 4-th step of the core equation, we have $g(\\mathbf{X}^t_{k+1})\\leq g(\\mathbf{X}^t_{k})\\leq...\\leq g(\\mathbf{X}^t_{0})$. Therefore, $\\mathrm{K}g(\\mathbf{X}^t_{\\mathrm{K}})\\leq\\sum_{k=1}^\\mathrm{K}g(\\mathbf{X}^t_{k})$ holds, which immediately follows by\\begin{equation} \\mathrm{K}g(\\mathbf{X}^t_\\mathrm{K})-\\mathrm{K}g(\\mathbf{X}^t_*)\\leq\\sum_{k=1}^\\mathrm{K} g(\\mathbf{X}^t_{k}) - \\mathrm{K}g(\\mathbf{X}^t_*)\\leq \\frac{1}{2\\eta}\\|\\mathbf{X}^t_0-\\mathbf{X}^t_*\\|^2\\leq\\frac{1}{2\\eta}\\epsilon^2. \\end{equation} Therefore, we have: \\begin{equation} g(\\mathbf{X}^t_\\mathrm{K})-g(\\mathbf{X}^t_*)\\leq \\frac{1}{2\\eta\\mathrm{K}}\\epsilon^2.\\end{equation}\n- On the approximate minimizer. We agree that the convergence justification relied on stringent assumptions (e.g., Gaussian kernel and separate convexity), and in practice only targets the approximate minimizer, which largely weakens its significance. In our revised manuscript, we will remove discussions on algorithmic convergence. Instead, we will limit the discussion in the appendix to the separate convexity of the objective under the linear kernel, thereby ensuring precision and avoiding overstatement.\n\n\n\n#### *[W2] The paper does not discuss the mechanism of missing data, limiting the scope of comparison with methods that perform well under MAR or MNAR.*\n**Response.** Thank you very much for your meticulous and actionable suggestion. In submission, relevant content was included in a **seperate supplementary appendix**, which may not have been sufficiently visible. To address this, we have now explicitly referenced the relevant discussions and results:\n- The MAR and MNAR mechanisms are discussed in `Appendix B.1`.\n- The MAR and MNAR results are discussed in `Appendix C.3` with missing ratio fixed as 0.1.\n\n#### *[W3] The method seems to be more of a completion strategy than a true imputation method. It is unclear how the method could be used to impute missing values on new data points without retraining the model, unlike other methods such as MICE or Miss.F. The experimental section should include a scenario with a held-out test dataset to evaluate generalization.*\n\n**Response.** Thank you for your insightful comment. Please see our concise response below.\n- Firstly, We agree that our method is best characterized as a **completion strategy**, similar to prior works [1,2]. To clarify this, we will revise the terminology throughout the manuscript to `completion` where appropriate. In particular, we will add the following statement to the preliminary note (line 58):\n> As a preliminary note, there are two primary paradigms for handling missing data: (1) completion, where the model is optimized to fill in missing entries within a given incomplete dataset and does not generalize to new missing indices; and (2) imputation, where the model is trained to impute missing values for previously unseen indices or samples. The completion setting is practical in contexts where the entire dataset is collected prior to analysis. Consistent with prior work such as Sinkhorn and TDM [1,2], this study adopts the completion paradigm.\n- Secondly, as more of a completion approach, the current method does not naturally generalize to imputing missing values in new data points without training. \n- Finally, we are pleased that **it is feasible to add the requested experiment using a held-out test dataset to evaluate how well KPI generalizes to new samples**. Specifically, for each new sample, KPI minimizes the learning objective (Eq. 8) via gradient descent, updating only the values on missing indices in the test sample while keeping those in the training data fixed. The results in the table below demonstrate that KPI retains effective in the held-out test setting.\n\n| Dataset  |CC| | IS| |\n|---|-|--|---|--|\n|    | MAE   | WASS  | MAE   | WASS  | \n| **MICE**     | 0.258 | 0.045     | 0.544 | 0.737     |\n| **MISS.F**     |0.222|0.040| 0.194 | 0.085     |\n| **KPI**  | 0.107 | 0.054     | 0.107 | 0.069     |\n\n**Reference**\n\n[1] Missing data imputation using optimal transport. ICML.\n\n[2] Transformed distribution matching for missing value imputation. ICML.\n\n#### *[W4] Notation is sometimes imprecise or not rigorous. For example, the function is applied to inputs that may contain Na values, which is not clearly defined or explained.*\n**Response.** Thank you for highlighting the notational clarity. In Eq. 4, we formulated $f^* (\\mathbf{X}^\\mathrm{(miss)}\\_{\\cdot,-d},\\mathbf{X}^\\mathrm{(obs)}\\_{\\cdot,-d})$. \nAs you noted, it is ambiguous to apply $f\n^*$ to inputs with NaN values. Nevertheless, in practice, as you could have noticed, **the missing entries are initialized with real values prior to optimization**, so $f$ only receives **real-valued inputs** during the KPI process. To avoid confusion, we will replace $\\mathbf{X}^\\mathrm{(miss)}$ with $\\mathbf{X}^\\mathrm{(comp)}$ —explicitly denoting completed values—in this and related expressions. Other notations will similarly be standarized to ensure rigor and consistency."}, {"review_id": "dmyV2ZALTz", "reviewer": "Reviewer_dCSM", "summary": "This paper proposes a kernel point imputation (KPI) framework for missing data imputation, aiming to address the issues of model misspecification and insufficient utilisation of oracle features in existing iterative imputation methods. The framework is implemented through a bi-level optimization : the inner layer adaptively selects the optimal model form for each feature in the reproducing kernel Hilbert space (RKHS) to mitigate model misspecification; while the outer layer uses oracle features as supervisory signals to optimise the imputation values. Experiments demonstrate that KPI outperforms existing methods on multiple real-world datasets and effectively leverages oracle features to enhance imputation accuracy.", "questions": "See Weaknesses and Questions.", "limitations": "n/a", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "**Advantages**\n\n1. Reformulates iterative imputation as a bi-level optimization problem, with the inner layer optimising the model form in the RKHS and the outer layer using oracle features to supervise the imputation. This provides a new paradigm for missing data imputation.\n\n2. The method is simple and easy to understand.\n\n3. Experimental results appear promising.\n\n**Weaknesses and Questions**\n\n1. The KPI framework proposed in the paper does not account for potential noise in the dataset, yet noise is prevalent in industrial scenarios and other real-world applications, which may limit KPI's applicability in such contexts.\n\n2. The paper defines oracle features as fully observed features. How can we determine which features can serve as oracle features in practical applications?\n\n3. The KPI method relies on oracle features. Can the KPI method perform effectively when there are few oracle features? In the experimental results shown in Figure 3, the KPI method is only compared with the Miss.F method. However, based on the experimental results in Table 1, the Miss.F method is not the suboptimal method.\n\n4. In the experimental results in Table 1, only the average results under different missing rates are shown, making it impossible to assess the effectiveness of the KPI method under high missing rates. It is recommended to present the results under different missing rate conditions separately to better validate the effectiveness of the KPI method.\n\n5. In the provided anonymous code link, there is only one file, and its content is ‘okasddsa’.\n\n**Summary**\n\nThis paper proposes a new paradigm by reformulating iterative imputation as a bi-level optimization problem. However, the KPI method heavily relies on oracle features, and the experiments conducted on different proportions of oracle features are not sufficiently comprehensive. Therefore, I recommend a borderline reject.", "was_revised": true, "final_justification": "Please see my original comments. I will stand my rating.", "rebuttal": "#### *[W1] The KPI framework proposed in the paper does not account for potential noise in the dataset, yet noise is prevalent in industrial scenarios and other real-world applications, which may limit KPI's applicability in such contexts.*\n**Response.** We appreciate the reviewer’s insightful observation regarding the role of noise in real-world datasets and its implications to KPI. We address this concern with a rigorous and evidence-based discussion as follows:\n- Firstly, as acknowledged in our limitations,  KPI assumes that observed data are accurate and free from significant noise. **Notably, that this assumption is not unique to KPI but is a foundational premise shared by most MDI methods in the literature**. We restate and emphasize this general assumption in the limitations section.\n- Secondly, the absence of explicit design for noisy data **does not necessarily imply poor robustness to noise**. To assess KPI under noisy conditions, we conducted additional experiments:\n  - Experimental protocol. We fixed the missing data ratio at 0.1 and introduced additive noise by randomly perturbing 10% of the observed entries, with each perturbed value altered by up to ±20% of its original magnitude. \n  - Experimental results. The results, summarized in the table below, demonstrate that KPI consistently achieves the best performance across noise-involved datasets. The results demonstrate that, although KPI is not specifically designed for noisy data, its superior performance is retained in the noisy scenario.\n\n\n| Dataset:| BT | CC | CBV | IS | PK | QB | WQW |\n|:-|-:|-:|-:|-:|-:|-:|-:|\n| CSDIT|0.6896 | 0.9024 |0.8501 | 0.7580 | 0.7001 |0.5579 | 0.7834 |\n| GAIN |0.7801 | 0.8263 |0.7957 | 0.6242 | 0.5258 |0.4736 | 0.7403 |\n| MICE |0.8743 | 1.0403 |1.0521 | 0.9135 | 0.7402 |0.8541 | 1.0602 |\n| MIRACLE|0.5587 | 0.6591 |0.7735 | 3.7138 | 3.8631 |0.6423 | 0.5978 |\n| MISS.F |0.4886 | 0.7359 |0.7267 | 0.5802 | 0.5919 |0.4346 | 0.6718 |\n| MIWAE|0.5782 | 0.7329 |0.8604 | 0.6242 | 0.6107 |0.5308 | 0.6985 |\n| Sinkhorn |0.5260 | 0.4990 |0.5549 | 0.5043 | 0.4782 |nan| 0.6262 |\n| TDM|0.6394 | 0.7565 |0.7637 | 0.6191 | 0.6064 |0.5083 | 0.7354 |\n| NewImp|0.7802 | 0.8465 |0.8145 | 0.7479 | 0.7558 |0.6046 | 0.7649 |\n| KPI|**0.4446** | **0.3672** |**0.3982** | **0.4123** | **0.3401** |**0.2396** | **0.5112** |\n\n\n#### *[W2] The paper defines oracle features as fully observed features. How can we determine which features can serve as oracle features in practical applications?*\n**Response.** Thank you for raising this important point regarding the practical identification of oracle features. In our framework, \"oracle features\" are operationally defined as those features that are fully observed—i.e., features for which no entries are missing across all samples in the dataset. **This definition is directly verifiable in real-world data.**\n\n- In practical applications, the selection of oracle features is typically straightforward: **one examines the data matrix and identifies features (columns) with complete observations.** For instance, in electronic health records, demographic variables such as `age`, `gender`, and `height` are routinely and reliably collected for all patients, and thus can be designated as oracle features. In industrial monitoring scenarios, sensor readings from robust or easily accessible equipment (e.g., ambient temperature, pressure at main control points) are often fully recorded, whereas measurements from sensors in harsh or inaccessible environments (e.g., temperature at the base of a high-pressure tower) are more prone to missingness due to equipment failure or maintenance constraints. In such cases, the consistently observed features naturally serve as oracle features.\n\n- We recognize that the number of fully observed features may be limited in some domains. However, our empirical results (Section 4.3) show that even a small number of oracle features can suffice for effective application of KPI. Even worse, in cases where no feature is entirely observed, the definition of \"oracle features\" may be relaxed to include those with very few missingness, which can be solved through suitable imputation prior to KPI process. We will clarify this operational flexibility and its implications in the revised manuscript.\n\n\n#### *[W3] The KPI method relies on oracle features. Can the KPI method perform effectively when there are few oracle features? In the experimental results shown in Figure 3, the KPI method is only compared with the Miss.F method. However, based on the experimental results in Table 1, the Miss.F method is not the suboptimal method.*\n\n**Response.** Thank you for your valuable and actionable comment. We agree that it is important to evaluate our method in the scenario where only a few oracle features are available and to include additional baselines.\n\n**Initially, we selected Miss.F for comparison because it is a strong representative within iterative imputation methods**, despite not being the overall best performer in Table 1. To address your concern, we have **added experiments using NewImp as a new baseline and investigated the case where only a small number of oracle features are accessible.**\n- Experimental protocol. We simulate this setting using CC and IS datasets, varying the number of oracle features (1, 2, 3, 4), where oracles are specified in order from left to right. Missingness is then introduced to the other features, with a fixed missing ratio of 0.1.\n- Experimental results. The table below summarizes the results for each method under varying numbers of oracle features. As the number of oracle features increases, the performance of KPI improves noticeably. These findings demonstrate that KPI is able to leverage the information from oracle features effectively, leading to substantial performance gains as more fully observed features are available.\n\n| Number of oracle features  | 1 |  | 2 |  | 3 |  | 4 |  |\n|---|--|--|--|--|--|--|--|--|\n|  | MAE  | WASS | MAE  | WASS | MAE  | WASS | MAE  | WASS |\n|CC dataset|\n| MISS.F  | 0.820  | 0.231  | 0.648  | 0.149  | 0.646  | 0.149  | 0.644  | 0.147  |\n| NewImp  | 0.620  | 0.130  | 0.530  | 0.115  | 0.560  | 0.120  | 0.540  | 0.112  |\n| KPI| **0.387**  | **0.083**  | **0.370**  | **0.079**  | **0.369**  | **0.074**  | **0.206**  | **0.022**  |\n| IS dataset|\n| MISS.F  | 0.419  | 0.459  | 0.328  | 0.191  | 0.330  | 0.233  | 0.338  | 0.228  |\n| NewImp  | 0.415  | 0.445  | 0.340  | 0.175  | 0.335  | 0.220  | 0.325  | 0.200  |\n| KPI | **0.363** | **0.426**| **0.268** | **0.114**| **0.268**  |  **0.102**| **0.155**  |  **0.061**|\n\n#### *[W4] In Table 1, only the average results under different missing rates are shown, making it impossible to assess the effectiveness of the KPI method under high missing rates. It is recommended to **present the results under different missing rates separately** to better validate the effectiveness of the KPI method.*\n\n**Response.** Thank you very much for your meticulous and actionable suggestion. In submission, relevant content was included in a **seperate supplementary appendix**, which may not have been sufficiently visible. To address this, we have now explicitly referenced the relevant discussions and results:\n- The results under different missing ratios are seperately reported in `Appendix C.2` (Table 5-8 for missing ratio 0.1, 0.2, 0.3, 0.4).\n- We further provide the results given higher missing ratio (0.6) as follow.\n\n|Datasets | BT  | | CC  | | CBV | | IS  |  | PS  |  | QB  |  | WQW |  |\n|-|-|-|-|-|-|-|-|--|-|--|-|--|-|--|\n| | MAE | WASS  | MAE | WASS  | MAE | WASS  | MAE | WASS | MAE | WASS | MAE | WASS | MAE | WASS \nCSDI-T | 0.746 | 1.647 | 0.931 | 3.045 | 0.843 | 4.111 | 0.772 | 15.419 | 0.721 | 10.429 | 0.558 | 21.528 | 0.754 | 4.54 \nMiss.D | 0.783 | 1.746 | 1.006 | 3.32  | 0.877 | 4.399 | 0.841 | 16.961 | 0.743 | 11.264 | 0.591 | 22.604 | 0.769 | 4.858  \nMICE | 0.744 | 0.231 | 1.009 | 0.983 | 0.987 | 1.785 | 0.79  | 11.091 | 0.531 | 3.224  | 0.698 | 10.544 | 0.955 | 1.684  \nGAIN | 0.727 | 0.72  | 1.016 | 2.233 | 1.294 | 5.083 | 1.26  | 24.406 | 0.799 | 7.758  | 1.188 | 53.39  | 1.423 | 7.177  \nSinkhorn | 0.655 | 0.696 | 0.766 | 1.317 | 0.742 | 1.782 | 0.587 | 7.577  | 0.539 | 4.601  | 0.435 | 8.563  | 0.683 | 1.848  \nTDM  | 0.736 | 0.766 | 0.81  | 1.426 | 0.808 | 2.221 | 0.714 | 9.394  | 0.692 | 6.313  | 0.562 | 10.234 | 0.75  | 2.199  \nMIWAE  | 0.712 | 0.631 | 0.828 | 1.244 | 0.861 | 2.166 | 0.691 | 9.833  | 0.595 | 4.63 | 0.521 | 12.082 | 0.712 | 1.615  \nMISS.F | 0.649 | 0.505 | 0.832 | 1.467 | 0.837 | 2.054 | 0.758 | 11.139 | 0.58  | 3.886  | 0.491 | 8.365  | 0.768 | 2.063  \nRemasker | **0.503** | 0.503 | 1.898 | 4.303 | 0.732 | 2.427 | 1.328 | 19.88  | 2.761 | 26.829 | 4.265 | 82.597 | **0.537** | **1.455**  \nNewImp | 0.75  | 0.807 | 0.826 | 1.491 | 0.818 | 2.317 | 0.759 | 9.904  | 0.75  | 6.624  | 0.6 | 11.45  | 0.765 | 2.264  \nKPI  | 0.598 | **0.435** | **0.557** | **0.841** | **0.732** | **1.585** | **0.517** | **6.835**  | **0.465** | **3.097**  | **0.394** | **7.803**  | 0.708 | 1.533  \n\n#### *[W5] In the provided anonymous code link, there is only one file, and its content is ‘okasddsa’.*\n\n**Response.** We sincerely apologize for the previous oversight regarding the anonymous code repository. The issue arose from a miscommunication within our team, which has now been thoroughly addressed. We have since updated the repository to include the complete, fully reproducible implementation of the KPI method, including all necessary scripts for data preprocessing, model training, and evaluation. Additionally, we have included a comprehensive README file that provides detailed instructions on the installation of dependencies and execution of experiments. Once again, we apologize for inconvenience this may have caused."}, {"review_id": "XZXGMogOjd", "reviewer": "Reviewer_8EZS", "summary": "This paper addresses two critical defects in existing iterative imputation methods for handling missing data: model misspecification (i.e., applying a uniform model form to heterogeneous data) and underutilization of \"oracle features\" (fully observed features) It proposes a bi-level optimization framework called Kernel Point Imputation (KPI) to solve these problems.", "questions": "1. Does the paper provide more detailed guidelines for identifying and defining \"oracle\" features? In practical applications, how would an inappropriate selection of \"oracle\" features impact imputation performance?\n\n2. What are the detailed implementation specifics of the adaptive kernel ensemble strategy? How does it ensure dynamic selection of the optimal kernel during training while avoiding overfitting?", "limitations": "The applicability of the model to missing imputation in non-tabular data (e.g., time series data) is not yet clear.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. Proposes a bi-level optimization framework that cleverly addresses the two core issues in iterative imputation: model misspecification and underutilization of \"oracle features.\"\n\n2. By adaptively selecting function forms in RKHS, KPI can better capture complex heterogeneous dependencies among different features.\n\nWeaknesses:\n\n1. The baselines compared in this paper are too outdated; at least some more competitive baselines published in 2024 should be added. Such as:\n\n[1]Jolicoeur-Martineau, Alexia, Kilian Fatras, and Tal Kachman. \"Generating and imputing tabular data via diffusion and flow-based gradient-boosted trees.\" In International Conference on Artificial Intelligence and Statistics, pp. 1288-1296. PMLR, 2024.\n[2]Wei, T. R., Wang, Y., Inoue, Y., Wu, H. T., & Fang, Y. (2024). Table Transformers for imputing textual attributes. Pattern Recognition Letters, 186, 258-264.\n[3]Liu, Yixin, Thalaiyasingam Ajanthan, Hisham Husain, and Vu Nguyen. \"Self-supervision improves diffusion models for tabular data imputation.\" In Proceedings of the 33rd ACM International Conference on Information and Knowledge Management, pp. 1513-1522. 2024.\n\n2. Computational Cost: The bi-level optimization framework and the use of RKHS might increase computational complexity, especially when dealing with large-scale datasets, potentially making it less efficient than some simpler imputation methods.\n\n3. Despite the adaptive kernel ensemble, the choice and parameterization of kernel functions might still impact performance and require careful tuning.\n\n4. Definition and Availability of \"Oracle\" Features: The assumption of \"oracle features\" might be difficult to meet or clearly define in some real-world scenarios, which could limit the method's universality.", "was_revised": true, "final_justification": "Thank you for the authors’ response. After carefully reviewing the reply, I have decided to retain my current score.", "rebuttal": "Thank you very much for your encouraging support and appreciation of our framework and addressed research problems. Below are our responses to the specific query raised.\n\n---\n\n#### *[W1] The baselines compared in this paper are too outdated; at least some more competitive **baselines published in 2024** should be added.*\n**Response.** We appreciate your insightful comment regarding the inclusion of more recent and competitive baselines. We agree that it is essential to ensure up-to-date comparison.\n\n- We would like to clarify that our current manuscript already incorporates **NewImp (NeurIPS 2024) and Remasker (ICLR 2024)**, both of which represent state-of-the-art methods published in 2024. Their inclusion represents the recent advances in the field and could contemporarily alleviate the concern regarding baseline selection.\n\n- In response to the suggestion, we are actively working to implement them in our experimental framework. Given the technical complexity of diffusion models that we are not familiar with, we respectfully request additional time to implement them. We will report the results of these new baselines in the revised manuscript or, if the review permits this, in a subsequent discussion phase.\n\n\n#### *[W2] Computational Cost: The bi-level optimization framework and the use of RKHS might increase computational complexity, especially when dealing with large-scale datasets, potentially making it less efficient than some simpler imputation methods.*\nYes, we acknowledge that computational cost is a key consideration when evaluating the proposed method. We address this concern with a rigorous and evidence-based discussion as follows:\n- Firstly, although KPI targets to solve a bi-level optimization problem, the final solution reduces to **a standard quadratic loss function (see Eq. 8) that is directly computable**. Similar to standard deep learning methods, KPI updates the loss function in a batch-wise manner without requiring bi-level optimization in practice, and the batch-wise update makes it scalable to large-scale datasets.\n- Secondly, **we report below the computational cost of KPI under various settings.** As complexity is primarily determined by the kernel matrix size—which is equal to the batch size—we focus on the running time of computing the learning objective (Eq. 8) per batch across different batch sizes. Results show that while larger batch sizes increase computational cost, **the running time remains limited ($<3.295$ ms given batch size 1024)**. We also showcase the complexity is less relevant to the feature dimension, since it does not change the kernel matrix size.\n\n| Batch size        | 32    | 64    | 256   | 512   | 1024   |\n|-------------------|-------|-------|-------|-------|--------|\n| Gaussian kernel   | 2.121 | 2.061 | 2.114 | 2.623 | 3.295  |\n| Polynomial kernel | 1.378 | 1.301 | 1.381 | 1.859 | 3.041  |\n\n| Feature number    | 16    | 32    | 64    | 128   | 256    |\n|-------------------|-------|-------|-------|-------|--------|\n| Gaussian kernel   | 2.064 | 2.062 | 2.064 | 2.068 | 2.223  |\n| Polynomial kernel | 1.307 | 1.311 | 1.329 | 1.311 | 1.302  |\n\n\n#### *[W3] Despite the adaptive kernel ensemble, the choice and parameterization of kernel functions might still impact performance and require careful tuning.*\n**Response.** Thank you for highlighting the important issue of kernel selection and parameterization. To address this concern, we demonstrate that Gaussian kernel is a good default choice prior to selection. Our response contains theoretical justification and empirical evidence.\n\n- **Theoretically**, the Gaussian kernel is known for its universality, allowing it to approximate any continuous function in RKHS. This property is essential for our method, as it ensures sufficient expressiveness to represen $f^*$ with kernel matrices (Lemma 3.3). Thus, the universality property makes the Gaussian kernel a strong default choice.\n- **Empirically**, we compare the Gaussian kernel to other commonly used kernels, such as linear, polynomial, and Laplacian. As shown in the table below, the Gaussian kernel consistently provides competitive or superior performance.\n- In conclusion, **Gaussian kernel serves as a theoretically and empirically advantageous choice, reducing the need for kernel selection**. Its sole hyperparameter, the kernel width σ, can be efficiently addressed using a multi-kernel ensemble strategy that adaptively selects and weights kernels with different σ values.\n\n| Kernel    | MSE   | WASS  | MAE   |\n|-----------|-------|-------|-------|\n|CC|\n| Linear    | 0.099 | 0.051 | 0.203 |\n| Poly      | **0.065** | 0.039 | 0.091 |\n| Laplacian | 0.068 | 0.042 | 0.093 |\n| Gaussian  | 0.076 | **0.035** | **0.082** |\n|CBV|\n| Linear    | 0.089 | 0.087 | 0.219 |\n| Poly      | 0.087 | 0.088 | 0.214 |\n| Laplacian | **0.083** | 0.080 | 0.211 |\n| Gaussian  | 0.087 | **0.066** | **0.205** |\n|BT|\n| Linear    | 0.326 | 0.101 | 0.359 |\n| Poly      | 0.305 | 0.090 | 0.342 |\n| Laplacian | 0.316 | 0.091 | 0.346 |\n| Gaussian  | **0.302** | **0.089** | **0.338** |\n\n\n\n#### *[W4] Definition and Availability of \"Oracle\" Features: The assumption of \"oracle features\" might be difficult to meet or clearly define in some real-world scenarios, which could limit the method's universality.*\n**Response.** Once again, thank you very much for your constructive comment. In our work, \"oracle features\" are defined as features that are fully observed, i.e., with no missing entries across all samples.\n\n- **Oracle features are ambiguous in real-world datasets:**  For example, in electronic health records, demographic variables such as age, gender, and height are typically fully observed. Similarly, in industrial settings, sensor readings from robust or easily accessible equipment (e.g., ambient temperature, pressure at main control points) are often fully recorded, whereas measurements from sensors in harsh environments (e.g., temperature at the base of a high-pressure tower) are more prone to missingness due to equipment failure or maintenance constraints. \n\n- **A few oracle features improve performance**.  We recognize that the number of fully observed features may be limited in some domains. However, our empirical results (Section 4.3) show that even a small number of oracle features can suffice for effective application of KPI.\n\n- **Relaxed definition.** In cases where no feature is entirely observed, the definition of \"oracle features\" may be relaxed to include those with very minimal missingness, after suitable imputation prior to KPI process. We will clarify this operational flexibility and its implications in the revised manuscript."}, {"review_id": "znQUvA7Ky7", "reviewer": "Reviewer_is18", "summary": "This paper presents Kernel Point Imputation (KPI), a novel method that addresses key weaknesses in iterative imputation. It uses a bi-level optimization framework to tackle model misspecification by adapting model forms within an RKHS and leverages fully observed oracle features for supervision.", "questions": "- The word oracle feature is not properly defined. Are they just observed features? Then the word \"oracle\" seems wordy. Or is it defined by the features with minimal missing values, as in line 94? Also, in line 94, the phrase \"minimal\" seems unclear.\n- How does the computational cost of KPI scale compared to baselines, given the expensive kernel matrix inversion? No running time is reported. The dataset's details and relevant data citations are not given. So, it's not certain that the method can be scaled to big datasets.\n- The theoretical results should be placed in the main text, while the proofs can be given in the appendix.\n- the missing rate explored are between 0.1-0.4, lacks of analysis for high missing percentages.\n- regarding noise in datasets as in line 305, it has been studied that missing data imputation can have a denoising property, at least for time series, as in\nMissing data imputation for noisy time-series data and applications in healthcare\nLP Le, XHN Thi, T Nguyen, MA Riegler, P Halvorsen, BT Nguyen\narXiv preprint arXiv:2412.11164\n- the related works should be improved", "limitations": "NA. \nthe author says that \"There is no societal impact of the work performed.\"", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\nThe method's key strengths are its innovative bi-level optimization framework, strong empirical results outperforming established baselines, and a principled method for leveraging oracle features as supervision, which existing methods often neglect. The result is fair. The algorithm is supported by theoretical justification\n\nWeaknesses:\nThe primary weaknesses are potential scalability issues due to computationally expensive kernel matrix inversion and the use of a heuristic kernel ensemble strategy.  \nexperiments details lacked", "was_revised": true, "final_justification": "The authors have addressed my concern very thoroughly with promising newly updated results. So, I recommend accepting the paper.", "rebuttal": "Thank you very much for your encouraging support and appreciation of our empirical results and theoretical justification. Below are our responses to the specific query raised.\n\nWe observed that some comments in the weaknesses and questions sections convey similar points. We will selectively merge them to make the response concise and easy to follow.\n\n\n---\n\n#### *[Q1] The word oracle feature is not properly defined. Are they just observed features? Then the word \"oracle\" seems wordy. Or is it defined by the features with minimal missing values, as in line 94? Also, in line 94, the phrase \"minimal\" seems unclear.*\n**Response.** Thank you very much for your concrete and meticulous comment. \n- Your understanding is absolutely correct: in our work, \"oracle features\" are defined as features that are fully observed, i.e., with no missing entries across all samples. We agree that the word \"oracle\" is wordy and will **replace `oracle features` with `fully observed features` in the revised manuscript**.\n- Moving forward, we understand that despite ambiguity of fully observed features, there are some scenarios without fully observed features. In this case, the scope of \"oracle features\" may be relaxed to include those features with very few (minimal) missing entries, which can be solved through suitable imputation prior to KPI process. We will discern this operational flexibility from the formal definition in the revised manuscript.\n\n\n#### *[Q2,W1] How does the computational cost of KPI scale compared to baselines, given the expensive kernel matrix inversion? No running time is reported. The dataset's details and relevant data citations are not given. So, it's not certain that the method can be scaled to big datasets.*\n\n**Response.** We express our gratitude for your meticulous and actionable feedback. There are two concerns in this query and we address them as follows:\n- **Dataset details and citation.** The datasets we use are subsets of varying scales from the UCI repository. We will provide detailed descriptions and proper citations for these datasets [1] in the revised manuscript.\n- **Computational cost and scalability concern.** \n  - KPI updates imputation values by minimizing the loss function (Eq. 8) in a batch-wise manner. Therefore, the complexity is determined by the batch size (which defines the kernel matrix size), rather than by the total number of samples, which enables scalability to large-scale datasets. \n  - On the basis, **we add experiments to investigate the running time per iteration given different batch size.** The results in the table below indicate that the running time remains under 3.**3.295 ms for a batch size of 1024**, demonstrating KPI’s efficiency. Meanwhile, the table demonstrates that the **complexity is less relevant to the feature number**, since altering the number of features does not affect the kernel matrix size.\n  - Collectively, **KPI's computational cost per iteration is primarily related to batch size and not to dataset size or feature dimensionality, making it scalable to large-scale datasets.**\n\n| Batch size  | 32  | 64  | 256 | 512 | 1024 |\n|----|-|-|-|-|--|\n| Gaussian kernel | 2.121 | 2.061 | 2.114 | 2.623 | 3.295  |\n| Polynomial kernel | 1.378 | 1.301 | 1.381 | 1.859 | 3.041  |\n\n| Feature number  | 16  | 32  | 64  | 128 | 256  |\n|----|-|-|-|-|--|\n| Gaussian kernel | 2.064 | 2.062 | 2.064 | 2.068 | 2.223  |\n| Polynomial kernel | 1.307 | 1.311 | 1.329 | 1.311 | 1.302  |\n\n\n\n#### *[Q3] The theoretical results should be placed in the main text, while the proofs can be given in the appendix.*\n**Response.** Thank you very much for your thoughtful suggestion. We fully agree that presenting the main theoretical results in the main text will enhance the clarity and impact of our manuscript. To address this, **we will explicitly include the key theorems at the end of Section 3.4**, ensuring that readers can readily access the central theoretical contributions without referring to the appendix. The detailed proofs will be provided in the appendix for completeness. We will also revise the \"Theoretical justification\" paragraph (lines 192-194) to reflect these changes and to clearly highlight the implications of our theoretical findings. We appreciate your guidance and are committed to making these improvements in the revised manuscript.\n\n\n#### *[Q4] The missing rate explored are between 0.1-0.4, lacks of analysis for high missing percentages.*\n**Response.** We express our gratitude for your constructive and actionable comment. **We have added the results for a high missing ratio of 0.6 in the table below.** Overall, KPI continues to outperform baseline methods even under this more challenging condition, highlighting its robustness to high missingness. We sincerely appreciate your valuable feedback and will update the manuscript accordingly to provide a more comprehensive evaluation.\n\n|Datasets | BT  | | CC  | | CBV | | IS  |  | PS  |  | QB  |  | WQW |  |\n|-|-|-|-|-|-|-|-|--|-|--|-|--|-|--|\n| | MAE | WASS  | MAE | WASS  | MAE | WASS  | MAE | WASS | MAE | WASS | MAE | WASS | MAE | WASS \nCSDI-T | 0.746 | 1.647 | 0.931 | 3.045 | 0.843 | 4.111 | 0.772 | 15.419 | 0.721 | 10.429 | 0.558 | 21.528 | 0.754 | 4.54 \nMiss.D | 0.783 | 1.746 | 1.006 | 3.32  | 0.877 | 4.399 | 0.841 | 16.961 | 0.743 | 11.264 | 0.591 | 22.604 | 0.769 | 4.858  \nMICE | 0.744 | 0.231 | 1.009 | 0.983 | 0.987 | 1.785 | 0.79  | 11.091 | 0.531 | 3.224  | 0.698 | 10.544 | 0.955 | 1.684  \nGAIN | 0.727 | 0.72  | 1.016 | 2.233 | 1.294 | 5.083 | 1.26  | 24.406 | 0.799 | 7.758  | 1.188 | 53.39  | 1.423 | 7.177  \nSinkhorn | 0.655 | 0.696 | 0.766 | 1.317 | 0.742 | 1.782 | 0.587 | 7.577  | 0.539 | 4.601  | 0.435 | 8.563  | 0.683 | 1.848  \nTDM  | 0.736 | 0.766 | 0.81  | 1.426 | 0.808 | 2.221 | 0.714 | 9.394  | 0.692 | 6.313  | 0.562 | 10.234 | 0.75  | 2.199  \nMIWAE  | 0.712 | 0.631 | 0.828 | 1.244 | 0.861 | 2.166 | 0.691 | 9.833  | 0.595 | 4.63 | 0.521 | 12.082 | 0.712 | 1.615  \nMISS.F | 0.649 | 0.505 | 0.832 | 1.467 | 0.837 | 2.054 | 0.758 | 11.139 | 0.58  | 3.886  | 0.491 | 8.365  | 0.768 | 2.063  \nRemasker | **0.503** | 0.503 | 1.898 | 4.303 | 0.732 | 2.427 | 1.328 | 19.88  | 2.761 | 26.829 | 4.265 | 82.597 | **0.537** | **1.455**  \nNewImp | 0.75  | 0.807 | 0.826 | 1.491 | 0.818 | 2.317 | 0.759 | 9.904  | 0.75  | 6.624  | 0.6 | 11.45  | 0.765 | 2.264  \nKPI  | 0.598 | **0.435** | **0.557** | **0.841** | **0.732** | **1.585** | **0.517** | **6.835**  | **0.465** | **3.097**  | **0.394** | **7.803**  | 0.708 | 1.533  \n\n#### *[Q5] Regarding noise in datasets as in line 305, it has been studied that missing data imputation can have a denoising property, at least for time series, as in Missing data imputation for noisy time-series data and applications in healthcare.*\n\n**Response.** The listed reference is inspiring to investigate the MDI problem in the presence of noise. Thank you for highlighting this important work. **We will add the related works in the revised manuscript.**\n- Currently, despite that KPI does not explicitly design for noisy data, it would be helpful to investigate KPI’s robustness to noise. To this end, **we add controlled experiments** by introducing additive noise (±20% perturbation to 10% of observed entries) at a fixed missing rate of 0.1. \n- **As shown in the table below, KPI consistently outperforms all baselines across diverse datasets under these noisy conditions**. These results empirically substantiate that, although KPI is not explicitly designed for denoising, it remains applicable to noisy case. We will incorporate a discussion of these findings and the relevant literature in the revised manuscript to provide a comprehensive and academically grounded perspective.\n\n| Dataset:| BT | CC | CBV | IS | PK | QB | WQW |\n|:-|-:|-:|-:|-:|-:|-:|-:|\n| CSDI-T|0.6896 | 0.9024 |0.8501 | 0.7580 | 0.7001 |0.5579 | 0.7834 |\n| GAIN |0.7801 | 0.8263 |0.7957 | 0.6242 | 0.5258 |0.4736 | 0.7403 |\n| MICE |0.8743 | 1.0403 |1.0521 | 0.9135 | 0.7402 |0.8541 | 1.0602 |\n| MIRACLE|0.5587 | 0.6591 |0.7735 | 3.7138 | 3.8631 |0.6423 | 0.5978 |\n| MISS.F |0.4886 | 0.7359 |0.7267 | 0.5802 | 0.5919 |0.4346 | 0.6718 |\n| MIWAE|0.5782 | 0.7329 |0.8604 | 0.6242 | 0.6107 |0.5308 | 0.6985 |\n| Sinkhorn |0.5260 | 0.4990 |0.5549 | 0.5043 | 0.4782 |Nan| 0.6262 |\n| TDM|0.6394 | 0.7565 |0.7637 | 0.6191 | 0.6064 |0.5083 | 0.7354 |\n| NewImp|0.7802 | 0.8465 |0.8145 | 0.7479 | 0.7558 |0.6046 | 0.7649 |\n| KPI|**0.4446** | **0.3672** |**0.3982** | **0.4123** | **0.3401** |**0.2396** | **0.5112** |\n\n#### *[Q6] the related works should be improved*\n\n**Response.** We appreciate the reviewer’s suggestion to enhance the related works section. Currently we group the related MDI methods into two main groups: discriminative and generative approaches. Within the discriminative category, we further differentiate between iterative methods (which KPI aims to improve) and non-iterative methods (e.g., TDM). Building on this taxonomy, we promise to update the related works section in the revised manuscript to incorporate recent advances and ensure a more comprehensive and up-to-date coverage of the field.\n\n**Reference**\n\n[1] Asuncion, Arthur, and David Newman. \"UCI machine learning repository.\" Nov. 2007."}]

Author rebuttal:

Meta-review: This paper presents a new method that uses an RKHS and oracle features (features with no or few missing attributes) to perform imputation. The experiments are thorough and demonstrate the method's superiority in imputation tasks compared to other methods. The method is also well-principled and has some supporting theory.

Three of the four reviewers recommend or lean towards accept. The fourth reviewer (dCSM) leans toward reject. This reviewer seems to have two lingering concerns. The first is a concern is that the use of oracle features is potentially limiting. Initially, several other reviewers shared this concern. However, the authors subsequently demonstrated robustness of their method to this aspect. Their response ultimately satisfied the other reviewers as well as me. 

The second concern was a lack of initial availability of code. However, NeurIPS policy makes this clear that a paper cannot be rejected solely for this, except where the code distribution is central to the paper, which it is not in this case.

Given this, I recommend acceptance of the paper despite the fourth reviewer's concerns.
Decision: Accept (poster)