ResearchPod Summary
Modern data science pipelines often struggle with missing data, typically relying on either restrictive parametric models (like multivariate Normality) or non-convex heuristic methods (like deep learning or iterative regression). These approaches often fail to capture the true joint distribution of the data, leading to biased downstream analyses. The authors ask: can we reframe the imputation problem as a convex density-estimation task that ensures distributional fidelity while remaining computationally tractable?
PSD-Impute treats imputation as a marginal-matching problem under the Missing Completely At Random (MCAR) mechanism. By leveraging positive semi-definite (PSD) kernel densities, the authors construct a model where the joint density is represented by a kernel-based feature map. This formulation allows for closed-form marginals, which are essential for handling missing values. The training objective is transformed into a convex empirical risk minimization problem, which is solved using a structured Newton interior-point method. This approach ensures that the resulting density is coherent and can be used for both single and multiple imputation.
The authors demonstrate that their PSD-Impute model provides a principled, distribution-aware alternative to existing methods. Theoretically, they establish that the estimator is statistically consistent and achieves minimax-optimal excess risk, effectively bypassing the curse of dimensionality for sufficiently smooth probability densities. Empirically, the model shows competitive performance across a variety of synthetic and real-world datasets, outperforming or matching popular baselines in terms of distributional accuracy. The use of a convex objective ensures that the model avoids the instability and non-convergence issues often associated with non-convex deep generative models.
This work bridges the gap between kernel density estimation and missing-data methodology. By providing a convex, theoretically grounded framework, it offers practitioners a reliable way to impute missing values that preserves the multivariate structure of the data. This is particularly valuable for downstream tasks that require a coherent joint density rather than simple point estimates, such as likelihood-based inference or uncertainty quantification.
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