Conventional matrix completion methods exploit the underlying low-rank structure of a target matrix to recover its missing entries, but may be inadequate for capturing complex nonlinear dependencies. Deep neural networks have therefore been increasingly adopted for nonlinear matrix completion (NLMC). However, their high representational capacity, particularly in deep fully connected neural networks (FCNNs), can lead to over-fitting when training is restricted to incomplete observations. In this paper, we propose a Deep Neural Network with Nonsmooth Regularization (DNN-NSR) framework to address this issue by incorporating two complementary structural priors: sparsity of intermediate representations and low-rankness of layer-wise weight matrices. Specifically, the $\ell_1$-norm of hidden representations promotes sparse latent features, while the nuclear norm of weight matrices encourages low-rank layer-wise transformations. To optimize the resulting nonsmooth and nonconvex objective, we develop an extrapolated proximal gradient algorithm that employs soft-thresholding and singular-value thresholding to handle the two regularization terms, respectively. We further introduce a gradual regularization strategy that progressively increases the influence of these structural penalties during training. Under suitable assumptions, we establish convergence properties of the proposed algorithm toward a critical point of the training objective. The effectiveness of DNN-NSR is evaluated through extensive experiments on synthetic matrix recovery, single-image inpainting, and recommender systems using the MovieLens 100K and MovieLens 1M datasets. Comparisons with representative linear and nonlinear matrix completion methods demonstrate the potential of the proposed framework to improve recovery performance by jointly exploiting sparse latent representations and low-rank network transformations.
Handling missing values is a common challenge in biostatistical analyses, typically addressed by imputation methods. We propose a novel, fast, and easy-to-use imputation method called missing value imputation with adversarial random forests (MissARF), based on generative machine learning, that provides both single and multiple imputation. MissARF employs adversarial random forest (ARF) for density estimation and data synthesis. To impute a missing value of an observation, we condition on the non-missing values and sample from the estimated conditional distribution generated by ARF. Our experiments demonstrate that MissARF performs comparably to state-of-the-art single and multiple imputation methods in terms of imputation quality and fast runtime with no additional costs for multiple imputation.
Recent years have witnessed a growing interest for time series foundation models, with a strong emphasis on the forecasting task. Yet, the crucial task of out-of-domain imputation of missing values remains largely underexplored. We propose a first step to fill this gap by leveraging implicit neural representations (INRs). INRs model time series as continuous functions and naturally handle various missing data scenarios and sampling rates. While they have shown strong performance within specific distributions, they struggle under distribution shifts. To address this, we introduce MoTM (Mixture of Timeflow Models), a step toward a foundation model for time series imputation. Building on the idea that a new time series is a mixture of previously seen patterns, MoTM combines a basis of INRs, each trained independently on a distinct family of time series, with a ridge regressor that adapts to the observed context at inference. We demonstrate robust in-domain and out-of-domain generalization across diverse imputation scenarios (e.g., block and pointwise missingness, variable sampling rates), paving the way for adaptable foundation imputation models.
Clinical laboratory results are ubiquitous in any diagnosis making. Predicting abnormal values of not prescribed tests based on the results of performed tests looks intriguing, as it would be possible to make early diagnosis available to everyone. The special place is taken by the Common Blood Count (CBC) test, as it is the most widely used clinical procedure. Combining routine biochemical panels with CBC presents a set of test-value pairs that varies from patient to patient, or, in common settings, a table with missing values. Here we formulate a tabular modeling problem as a set translation problem where the source set comprises pairs of GPT-like label column embedding and its corresponding value while the target set consists of the same type embeddings only. The proposed approach can effectively deal with missing values without implicitly estimating them and bridges the world of LLM with the tabular domain. Applying this method to clinical laboratory data, we achieve an improvement up to 8% AUC for joint predictions of high uric acid, glucose, cholesterol, and low ferritin levels.
The increasing reliance on machine learning (ML) models for decision-making requires high-quality training data. However, access to real-world datasets is often restricted due to privacy concerns, proprietary restrictions, and incomplete data availability. As a result, synthetic data generation (SDG) has emerged as a viable alternative, enabling the creation of artificial datasets that preserve the statistical properties of real data while ensuring privacy compliance. Despite its advantages, synthetic data is often overly clean and lacks real-world imperfections, such as missing values, noise, outliers, and misclassified labels, which can significantly impact model generalization and robustness. To address this limitation, we introduce Pucktrick, a Python library designed to systematically contaminate synthetic datasets by introducing controlled errors. The library supports multiple error types, including missing data, noisy values, outliers, label misclassification, duplication, and class imbalance, offering a structured approach to evaluating ML model resilience under real-world data imperfections. Pucktrick provides two contamination modes: one for injecting errors into clean datasets and another for further corrupting already contaminated datasets. Through extensive experiments on real-world financial datasets, we evaluate the impact of systematic data contamination on model performance. Our findings demonstrate that ML models trained on contaminated synthetic data outperform those trained on purely synthetic, error-free data, particularly for tree-based and linear models such as SVMs and Extra Trees.
Cross-domain time series imputation is an underexplored data-centric research task that presents significant challenges, particularly when the target domain suffers from high missing rates and domain shifts in temporal dynamics. Existing time series imputation approaches primarily focus on the single-domain setting, which cannot effectively adapt to a new domain with domain shifts. Meanwhile, conventional domain adaptation techniques struggle with data incompleteness, as they typically assume the data from both source and target domains are fully observed to enable adaptation. For the problem of cross-domain time series imputation, missing values introduce high uncertainty that hinders distribution alignment, making existing adaptation strategies ineffective. Specifically, our proposed solution tackles this problem from three perspectives: (i) Data: We introduce a frequency-based time series interpolation strategy that integrates shared spectral components from both domains while retaining domain-specific temporal structures, constructing informative priors for imputation. (ii) Model: We design a diffusion-based imputation model that effectively learns domain-shared representations and captures domain-specific temporal dependencies with dedicated denoising networks. (iii) Algorithm: We further propose a cross-domain consistency alignment strategy that selectively regularizes output-level domain discrepancies, enabling effective knowledge transfer while preserving domain-specific characteristics. Extensive experiments on three real-world datasets demonstrate the superiority of our proposed approach. Our code implementation is available here.
Time series imputation models have traditionally been developed using complete datasets with artificial masking patterns to simulate missing values. However, in real-world infrastructure monitoring, practitioners often encounter datasets where large amounts of data are missing and follow complex, heterogeneous patterns. We introduce DIM-SUM, a preprocessing framework for training robust imputation models that bridges the gap between artificially masked training data and real missing patterns. DIM-SUM combines pattern clustering and adaptive masking strategies with theoretical learning guarantees to handle diverse missing patterns actually observed in the data. Through extensive experiments on over 2 billion readings from California water districts, electricity datasets, and benchmarks, we demonstrate that DIM-SUM outperforms traditional methods by reaching similar accuracy with lower processing time and significantly less training data. When compared against a large pre-trained model, DIM-SUM averages 2x higher accuracy with significantly less inference time.
The low-rank matrix completion problem asks whether a given real matrix with missing values can be completed so that the resulting matrix has low rank or is close to a low-rank matrix. The completed matrix is often required to satisfy additional structural constraints, such as positive semi-definiteness or a bounded infinity norm. The problem arises in various research fields, including machine learning, statistics, and theoretical computer science, and has broad real-world applications. This paper presents new $\mathsf{NP} $-hardness results for low-rank matrix completion problems. We show that for every sufficiently large integer $d$ and any real number $\varepsilon \in [ 2^{-O(d)},\frac{1}{7}]$, given a partial matrix $A$ with exposed values of magnitude at most $1$ that admits a positive semi-definite completion of rank $d$, it is $\mathsf{NP}$-hard to find a positive semi-definite matrix that agrees with each given value of $A$ up to an additive error of at most $\varepsilon$, even when the rank is allowed to exceed $d$ by a multiplicative factor of $O (\frac{1}{\varepsilon ^2 \cdot \log(1/\varepsilon)} )$. This strengthens a result of Hardt, Meka, Raghavendra, and Weitz (COLT, 2014), which applies to multiplicative factors smaller than $2$ and to $\varepsilon $ that decays polynomially in $d$. We establish similar $\mathsf{NP}$-hardness results for the case where the completed matrix is constrained to have a bounded infinity norm (rather than be positive semi-definite), for which all previous hardness results rely on complexity assumptions related to the Unique Games Conjecture. Our proofs involve a novel notion of nearly orthonormal representations of graphs, the concept of line digraphs, and bounds on the rank of perturbed identity matrices.
Decision trees are widely used for interpretable machine learning due to their clearly structured reasoning process. However, this structure belies a challenge we refer to as predictive equivalence: a given tree's decision boundary can be represented by many different decision trees. The presence of models with identical decision boundaries but different evaluation processes makes model selection challenging. The models will have different variable importance and behave differently in the presence of missing values, but most optimization procedures will arbitrarily choose one such model to return. We present a boolean logical representation of decision trees that does not exhibit predictive equivalence and is faithful to the underlying decision boundary. We apply our representation to several downstream machine learning tasks. Using our representation, we show that decision trees are surprisingly robust to test-time missingness of feature values; we address predictive equivalence's impact on quantifying variable importance; and we present an algorithm to optimize the cost of reaching predictions.
We present a unified framework for solving partial differential equations (PDEs) using video-inpainting diffusion transformer models. Unlike existing methods that devise specialized strategies for either forward or inverse problems under full or partial observation, our approach unifies these tasks under a single, flexible generative framework. Specifically, we recast PDE-solving as a generalized inpainting problem, e.g., treating forward prediction as inferring missing spatiotemporal information of future states from initial conditions. To this end, we design a transformer-based architecture that conditions on arbitrary patterns of known data to infer missing values across time and space. Our method proposes pixel-space video diffusion models for fine-grained, high-fidelity inpainting and conditioning, while enhancing computational efficiency through hierarchical modeling. Extensive experiments show that our video inpainting-based diffusion model offers an accurate and versatile solution across a wide range of PDEs and problem setups, outperforming state-of-the-art baselines.
In cognitive neuroscience research, Representational Dissimilarity Matrices (RDMs) are often incomplete because pairwise similarity judgments cannot always be exhaustively collected as the number of pairs rapidly increases with the number of conditions. Existing methods to fill these missing values, such as deep neural network imputation, are powerful but computationally demanding and relatively opaque. We introduce a simple algorithm based on geometric inference that fills missing dissimilarity matrix entries using known distances. We use tests on publicly available empirical cognitive neuroscience datasets, as well as simulations, to demonstrate the method's effectiveness and robustness across varying sparsity and matrix sizes. We have made this geometric reconstruction algorithm, implemented in Python and MATLAB, publicly available. This method provides a fast and accurate solution for completing partial dissimilarity matrices in the cognitive neurosciences.
This tutorial aims to provide signal processing (SP) and machine learning (ML) practitioners with vital tools, in an accessible way, to answer the question: How to deal with missing data? There are many strategies to handle incomplete signals. In this paper, we propose to group these strategies based on three common tasks: i) missing-data imputation, ii) estimation with missing values and iii) prediction with missing values. We focus on methodological and experimental results through specific case studies on real-world applications. Promising and future research directions, including a better integration of informative missingness, are also discussed. We hope that the proposed conceptual framework and the presentation of recent missing-data problems related will encourage researchers of the SP and ML communities to develop original methods and to efficiently deal with new applications involving missing data.
A unified foundation model for medical time series -- pretrained on open access and ethics board-approved medical corpora -- offers the potential to reduce annotation burdens, minimize model customization, and enable robust transfer across clinical institutions, modalities, and tasks, particularly in data-scarce or privacy-constrained environments. However, existing generalist time series foundation models struggle to handle medical time series data due to their inherent challenges, including irregular intervals, heterogeneous sampling rates, and frequent missing values. To address these challenges, we introduce MIRA, a unified foundation model specifically designed for medical time series forecasting. MIRA incorporates a Continuous-Time Rotary Positional Encoding that enables fine-grained modeling of variable time intervals, a frequency-specific mixture-of-experts layer that routes computation across latent frequency regimes to further promote temporal specialization, and a Continuous Dynamics Extrapolation Block based on Neural ODE that models the continuous trajectory of latent states, enabling accurate forecasting at arbitrary target timestamps. Pretrained on a large-scale and diverse medical corpus comprising over 454 billion time points collect from publicly available datasets, MIRA achieves reductions in forecasting errors by an average of 10% and 7% in out-of-distribution and in-distribution scenarios, respectively, when compared to other zero-shot and fine-tuned baselines. We also introduce a comprehensive benchmark spanning multiple downstream clinical tasks, establishing a foundation for future research in medical time series modeling.
Language models (LMs) are increasingly being deployed to perform autonomous data analyses. However, their data awareness -- the ability to recognize, reason over, and appropriately handle data artifacts such as missing values, outliers, and logical inconsistencies -- remains underexplored. These artifacts are especially common in real-world tabular data and, if mishandled, can significantly compromise the validity of analytical conclusions. To address this gap, we present RADAR, a benchmark for systematically evaluating data-aware reasoning on tabular data. We develop a framework to simulate data artifacts via programmatic perturbations to enable targeted evaluation of model behavior. RADAR comprises 2980 table query pairs, grounded in real-world data spanning 9 domains and 5 data artifact types. In addition to evaluating artifact handling, RADAR systematically varies table size to study how reasoning performance holds when increasing table size. Our evaluation reveals that, despite decent performance on tables without data artifacts, frontier models degrade significantly when data artifacts are introduced, exposing critical gaps in their capacity for robust, data-aware analysis. Designed to be flexible and extensible, RADAR supports diverse perturbation types and controllable table sizes, offering a valuable resource for advancing tabular reasoning.
Real-world time series data are inherently multivariate, often exhibiting complex inter-channel dependencies. Each channel is typically sampled at its own period and is prone to missing values due to various practical and operational constraints. These characteristics pose fundamental challenges related to channel dependency, sampling asynchrony, and missingness, all of which must be addressed to enable robust and reliable forecasting in practical settings. However, most existing architectures are built on oversimplified assumptions, such as identical sampling periods across channels and fully observed inputs at test time, which often do not hold in real-world scenarios. To bridge this gap, we propose ChannelTokenFormer, a Transformer-based forecasting model with a flexible architecture designed to explicitly capture cross-channel interactions, accommodate channel-wise asynchronous sampling, and effectively handle missing values. Extensive experiments on three benchmark datasets modified to reflect practical settings, along with one real-world industrial dataset, demonstrate the superior robustness and accuracy of ChannelTokenFormer under challenging real-world conditions.
This study investigates the application of machine learning (ML) models for classifying dental providers into two categories - standard rendering providers and safety net clinic (SNC) providers - using a 2018 dataset of 24,300 instances with 20 features. The dataset, characterized by high missing values (38.1%), includes service counts (preventive, treatment, exams), delivery systems (FFS, managed care), and beneficiary demographics. Feature ranking methods such as information gain, Gini index, and ANOVA were employed to identify critical predictors, revealing treatment-related metrics (TXMT_USER_CNT, TXMT_SVC_CNT) as top-ranked features. Twelve ML models, including k-Nearest Neighbors (kNN), Decision Trees, Support Vector Machines (SVM), Stochastic Gradient Descent (SGD), Random Forest, Neural Networks, and Gradient Boosting, were evaluated using 10-fold cross-validation. Classification accuracy was tested across incremental feature subsets derived from rankings. The Neural Network achieved the highest accuracy (94.1%) using all 20 features, followed by Gradient Boosting (93.2%) and Random Forest (93.0%). Models showed improved performance as more features were incorporated, with SGD and ensemble methods demonstrating robustness to missing data. Feature ranking highlighted the dominance of treatment service counts and annotation codes in distinguishing provider types, while demographic variables (AGE_GROUP, CALENDAR_YEAR) had minimal impact. The study underscores the importance of feature selection in enhancing model efficiency and accuracy, particularly in imbalanced healthcare datasets. These findings advocate for integrating feature-ranking techniques with advanced ML algorithms to optimize dental provider classification, enabling targeted resource allocation for underserved populations.
We present a new viewpoint on a reconstructing multidimensional geological fields from sparse observations. Drawing inspiration from deterministic image inpainting techniques, we model a partially observed spatial field as a multidimensional tensor and recover missing values by enforcing a global low-rank structure. Our approach combines ideas from tensor completion and geostatistics, providing a robust optimization framework. Experiments on synthetic geological fields demonstrate that used tensor completion method significant improvements in reconstruction accuracy over ordinary kriging for various percent of observed data.
Causal reasoning is essential for business process interventions and improvement, requiring a clear understanding of causal relationships among activity execution times in an event log. Recent work introduced a method for discovering causal process models but lacked the ability to capture alternating causal conditions across multiple variants. This raises the challenges of handling missing values and expressing the alternating conditions among log splits when blending traces with varying activities. We propose a novel method to unify multiple causal process variants into a consistent model that preserves the correctness of the original causal models, while explicitly representing their causal-flow alternations. The method is formally defined, proved, evaluated on three open and two proprietary datasets, and released as an open-source implementation.
We consider the problem of solving the optimal transport problem between two empirical distributions with missing values. Our main assumption is that the data is missing completely at random (MCAR), but we allow for heterogeneous missingness probabilities across features and across the two distributions. As a first contribution, we show that the Wasserstein distance between empirical Gaussian distributions and linear Monge maps between arbitrary distributions can be debiased without significantly affecting the sample complexity. Secondly, we show that entropic regularized optimal transport can be estimated efficiently and consistently using iterative singular value thresholding (ISVT). We propose a validation set-free hyperparameter selection strategy for ISVT that leverages our estimator of the Bures-Wasserstein distance, which could be of independent interest in general matrix completion problems. Finally, we validate our findings on a wide range of numerical applications.
Irregular Multivariate Time Series (IMTS) forecasting is challenging due to the unaligned nature of multi-channel signals and the prevalence of extensive missing data. Existing methods struggle to capture reliable temporal patterns from such data due to significant missing values. While pre-trained foundation models show potential for addressing these challenges, they are typically designed for Regularly Sampled Time Series (RTS). Motivated by the visual Mask AutoEncoder's (MAE) powerful capability for modeling sparse multi-channel information and its success in RTS forecasting, we propose VIMTS, a framework adapting Visual MAE for IMTS forecasting. To mitigate the effect of missing values, VIMTS first processes IMTS along the timeline into feature patches at equal intervals. These patches are then complemented using learned cross-channel dependencies. Then it leverages visual MAE's capability in handling sparse multichannel data for patch reconstruction, followed by a coarse-to-fine technique to generate precise predictions from focused contexts. In addition, we integrate self-supervised learning for improved IMTS modeling by adapting the visual MAE to IMTS data. Extensive experiments demonstrate VIMTS's superior performance and few-shot capability, advancing the application of visual foundation models in more general time series tasks. Our code is available at https://github.com/WHU-HZY/VIMTS.