Conformal Prediction with Cellwise Outliers: A Detect-then-Impute Approach

TL;DR

This work extends conformal prediction to settings with cellwise test-time outliers by introducing a detect-then-impute framework that restores exchangeability for calibrated prediction intervals. It develops two practical algorithms, PDI-CP and JDI-CP, and analyzes their distribution-free coverage properties, including a finite-sample $1-2\alpha$ guarantee for JDI-CP under a sure-detection assumption. The ODI-CP baseline establishes the theoretical benchmark, while PDI-CP and JDI-CP provide robust, flexible alternatives that wrap around common detection and imputation methods. Empirical results on synthetic and real data demonstrate robust coverage and competitive efficiency, with SCP and WCP failing under cellwise contamination. The approach offers a practical solution for reliable predictive uncertainty quantification in the presence of cellwise outliers across diverse domains.

Abstract

Conformal prediction is a powerful tool for constructing prediction intervals for black-box models, providing a finite sample coverage guarantee for exchangeable data. However, this exchangeability is compromised when some entries of the test feature are contaminated, such as in the case of cellwise outliers. To address this issue, this paper introduces a novel framework called detect-then-impute conformal prediction. This framework first employs an outlier detection procedure on the test feature and then utilizes an imputation method to fill in those cells identified as outliers. To quantify the uncertainty in the processed test feature, we adaptively apply the detection and imputation procedures to the calibration set, thereby constructing exchangeable features for the conformal prediction interval of the test label. We develop two practical algorithms, PDI-CP and JDI-CP, and provide a distribution-free coverage analysis under some commonly used detection and imputation procedures. Notably, JDI-CP achieves a finite sample $1-2α$ coverage guarantee. Numerical experiments on both synthetic and real datasets demonstrate that our proposed algorithms exhibit robust coverage properties and comparable efficiency to the oracle baseline.

Figures (15)

• Figure 1: An illustration of the relationship of ODI-CP, PDI-CP and JDI-CP. $\hat{{\mathcal{O}}}_i = \mathbf{D}(X_i)$ and $\tilde{{\mathcal{O}}}_{n+1} = \mathbf{D}(\tilde{X}_{n+1})$.
• Figure 2: An illustration of the similarity of $\tilde{{\mathcal{T}}}_{n+1}$ and $\hat{{\mathcal{T}}}_{n+1}$ (left axis, red cross marks) and the number of occurrences of empty and nonempty sets (right axis, blue and gray dots).
• Figure 3: Simulation results of six methods. $\mathbf{D}$ is DDC, $\mathbf{I}$ is Mean Imputation, $\epsilon = 0.1$ and $\tau_j=\sqrt{\chi _{1,0.95}^{2}}$. The red line is the target coverage level $1-\alpha = 90\%$.
• Figure 4: Simulation results of Baseline and our methods with different $\mathbf{D}$ procedures. $\mathbf{I}$ is Mean Imputation and $\epsilon = 0.1$.
• Figure 5: Simulation results of Baseline and our methods with different $\mathbf{I}$ procedures. $\mathbf{D}$ is DDC, $\epsilon = 0.1$ and $\tau_j=\sqrt{\chi_{1,0.95}^{2}}$.

Theorems & Definitions (18)

• Lemma 3.3
• Proposition 3.4
• Theorem 3.5
• Definition 4.1: $\ell_1$-sensitivity
• Definition 4.2: Mean Imputation
• Theorem 4.3
• Theorem 4.4
• Theorem 5.1
• Theorem 2.1
• proof