Sparse recovery in linear systems underpins applications from signal processing to high-dimensional regression. Sparse Bayesian Learning, grounded in the principle of automatic relevance determination (ARD), offers a practical Bayesian mechanism for feature sparsity via marginal likelihood optimization. Yet, its reliance on a homoscedastic noise model renders it sensitive to data contaminations such as outliers or misspecified noise, harming model fit and predictions. Instead, we propose jointly learning individual feature and sample relevancies, enabling simultaneous model and data sparsification via a single Bayesian objective. This symmetric pruning of model and data offers a natural extension that preserves conjugacy, admits closed-form updates for standard optimization procedures, and aligns with perspectives from robust regression and influence functions. Empirical results across diverse regression tasks affirm that a joint ARD approach consistently yields both sparse and robust prediction models.

We study the sample complexity of robust binary hypothesis testing under three standard contamination models: $\varepsilon$-additive (Huber), $\varepsilon$-subtractive, and $\varepsilon$-total variation (TV), denoted by $n^*_{\mathrm{Hub}}(\varepsilon)$, $n^*_{\mathrm{Sub}}(\varepsilon)$, and $n^*_{\mathrm{TV}}(\varepsilon)$, respectively. For subtractive contamination, we show that least favourable distributions exist and provide explicit formulas for the same, bringing this model in line with the classical Huber and TV models. Next we show that in all three models, sample complexity may be highly unstable in the contamination parameter $\varepsilon$, increasing by polynomial factors even for $o(\varepsilon)$ perturbations. Similarly, there may be polynomial factor gaps between the sample complexities when $\varepsilon$ is known exactly versus when it is known up to $o(\varepsilon)$ error. Despite the instability of the sample complexity in all models, we show that the sample complexities across models are comparable up to constant-factor rescaling of $\varepsilon$. Specifically, for any fixed $δ_0>0$, the following hold for all distributions $p$ and $q$: (i) $n^*_{\mathrm{Hub}}(\varepsilon) \lesssim n^*_{\mathrm{TV}}(\varepsilon) \lesssim n^*_{\mathrm{Hub}}(2\varepsilon)$, (ii) $n^*_{\mathrm{Sub}}(\varepsilon) \lesssim n^*_{\mathrm{TV}}(\varepsilon) \lesssim n^*_{\mathrm{Sub}}((2+δ_0)\varepsilon)$, and (iii) $n^*_{\mathrm{Sub}}(\varepsilon) \lesssim n^*_{\mathrm{Hub}}(\varepsilon) \lesssim n^*_{\mathrm{Sub}}((1+δ_0)\varepsilon)$, and the scaling constants are tight. Finally, we extend our results to adaptive versions of the contamination models.

Removing noise is difficult, but adding noise is easy. In this work, we show how to eliminate mean-shift noisy components from PCA by deliberately introducing knockoff mean-shift perturbation. Standard PCA is highly sensitive to shifts in the sample mean: a small fraction of samples from a shifted distribution can cause large deviations in the leading principal components. In high-dimensional regimes, existing Robust PCA approaches cannot handle the mean-shift contamination structure inherent in the mixture model. Using tools from Random Matrix Theory, we prove that the mean-shift spikes are spectrally separable from the stable eigenvalues of the original covariance. Furthermore, the original eigenspace remains asymptotically invariant to the contamination, independent of the mixture weight. Exploiting this spectral stability, we propose a simple, two-stage PCA algorithm by adding knockoff mean that identifies and removes the mean-shift component using only standard PCA operations.

Neural processes amortize Gaussian process inference, replacing the exact $O(n^3)$ posterior with a learned $O(n)$ map from context sets to predictive distributions. For a class of latent neural processes, we bound the Kullback--Leibler (KL) divergence between the GP and LNP predictives, decomposing it into three interpretable sources, namely label contamination as the neural process uses label values to estimate a quantity that is label-independent in the exact GP, an information bottleneck because the finite-dimensional representation cannot resolve the full context geometry, and amortization error from a single encoder network shared across all contexts. The bottleneck truncation term decays in the representation dimension $d$ as $O(e^{-cd^{2/d_x}})$ for squared-exponential kernels on $\mathbb{R}^{d_x}$ where $c > 0$ is a kernel-dependent constant and as $O(d^{-2ν/d_x})$ for Matérn-$ν$ kernels, directly linking architecture sizing to kernel smoothness and input dimension. The label contamination term is $O(1)$ in general, with only the observation-noise component decaying as $O(1/n)$, identifying a persistent cost of routing uncertainty estimation through a label-dependent representation. These results characterize the costs of amortization within the analyzed class and yield architectural recommendations to predict variance from context locations alone in the GP-amortization regime, and replace mean aggregation with second-order pooling to close the dominant amortization gap.

Trimming suspicious calibration points is a common response to contamination in conformal prediction. Its effect on clean-target coverage, however, is governed by the retained law induced by trimming, not by the contamination level alone. We analyse fixed-threshold trimming as conditioning rather than purification. It replaces the contaminated calibration law with a retained law, reducing clean-target coverage to a one-dimensional score-CDF transfer problem with an exact finite-sample identity. A componentwise bound on the transfer gap gives a population-level diagnostic. This separates a clean-side covariance cost from a retained-contamination cost, governed by the dirty-to-clean retention ratio. Trimming helps when the anomaly score separates retention probabilities while remaining score-neutral on the clean population. Otherwise, it cannot substantially reduce contamination through the retained mixture coefficient. We also give finite-sample certificate templates that provide numerical guarantees under independent audit.

Groundwater in the Densu Basin is increasingly threatened by heavy metal contamination, but conventional methods fail to capture the statistical complexity and spatial heterogeneity of pollution indicators. A key challenge is modelling the Heavy Metal Pollution Index (HPI), which is typically skewed and affected by correlated contaminants, leading to biased predictions without transformation. This study develops a predictive framework integrating response transformations with nested cross-validated ensemble machine learning. Three transformations (raw, log, and Gaussian copula) were applied to HPI and evaluated across six learners: support vector regression (SVM), $k$-nearest neighbours (k-NN), CART, Elastic Net, kernel ridge regression, and a stacked Lasso ensemble. Raw-scale models produced deceptively high fits (Elastic Net and stacked ensemble $R^2 \approx 1.0$), suggesting over-optimism. The log transformation stabilised variance (SVM: $R^2 = 0.93$, RMSE $= 0.18$; k-NN: $R^2 = 0.92$, RMSE $= 0.20$). The Gaussian copula gave the most reliable results: stacked ensemble $R^2 = 0.96$ (RMSE $= 0.19$), with other learners maintaining high accuracy. Copula-based models improved residuals and produced spatially plausible maps. DBSCAN clustering revealed Fe and Mn as primary HPI contributors, consistent with regional hydrogeochemistry. Limitations include reliance on random (not spatial) cross-validation and basin-specific scope. Future work should explore spatial validation and other geological settings. Overall, distribution-aware ensembles with clustering diagnostics offer robust, interpretable assessments of groundwater contamination.

Double-machine-learning pipelines for the Average Dose-Response Function rely on kernel-weighted local-linear smoothers, which inherit unbounded functional influence: a single outlier within a kernel window biases the curve across the entire window. We introduce SHIFT (Self-calibrated Heavy-tail Inlier-Fit with Tempering), a robust DML estimator combining cross-fit nuisance orthogonalization with a kernel-local Welsch-loss second stage optimized by Graduated Non-Convexity, and -- the principal design choice -- a defensive OLS refit whose inlier cutoff is scaled by post-GNC residual MAD rather than the raw-outcome MAD. On a localized-contamination stress test at $p=0.25$ this design choice drops level-RMSE from 1.03 to 0.33 while leaving clean and uniformly-contaminated runs unchanged. Across 1,400 main-sweep fits, SHIFT has competitive worst-case shape recovery (RMSE $0.325$ at $p=0.25$, second to Huber-DML's $0.276$); among the three methods with worst-case RMSE below $0.35$, only SHIFT emits a non-uniform per-sample weight vector, recovering the ground-truth outlier mask at mean $F_1 \approx 0.96$ (range $0.945$--$0.968$) on Gaussian-jump DGPs. We pair the estimator with a six-technique Extreme Value Theory diagnostic suite (Hill, GPD-MLE/PWM, GEV, Mean Excess, parameter stability, causal tail coefficient) that lets a practitioner distinguish Frechet from Weibull regimes and choose between SHIFT and L1 alternatives on empirical grounds. Extensions to binary-treatment CATE (Huber pseudo-outcome X-Learner) and time-series ADRF (block-CV + rolling MAD) are included. A counter-intuitive ablation: linear nuisance models (Ridge, Lasso) outperform gradient-boosted nuisances for robust DML under uniform contamination, inverting the usual more-flexible-is-better heuristic.

Conditional Average Treatment Effect (CATE) estimation in practice demands three properties simultaneously: heterogeneous effects τ(x), calibrated uncertainty over them, and robustness to the heavy tails that contaminate real outcome data. Meta-learners (Künzel et al., 2019) give (i); causal forests and BART give (i)-(ii) with Gaussian-tail assumptions; no widely used tool gives all three. We present Bayesian X-Learner, an X-Learner built on cross-fitted doubly robust pseudo-outcomes (Kennedy, 2020) with a full MCMC posterior over τ(x) via a Welsch redescending pseudo-likelihood. On Hill's IHDP benchmark the default configuration attains mean εPEHE = 0.56 on 5 replications (lowest mean; differences from S-/T-/X-learners, full-config Causal BART, and a causal forest baseline are not significant at α = 0.05, and rank ordering is unstable at 10 replications -- IHDP comparisons are competitive rather than dominant). On contaminated "whale" DGPs with up to 20-25% tail density, a one-flag extension (contamination_severity) that selects a Huberδ nuisance loss per Huber's minimax-δ relation recovers RMSE 0.13 with tight credible intervals (single-cross-fit 30-seed coverage 83% [Wilson 66%, 93%] at 20% density; modularBayes pooling with Bayesian-bootstrap nuisance draws restores nominal 95% coverage). We validate on the Hillstrom email-marketing RCT (N = 42,613), demonstrating consistent behaviour on real heavy-tailed outcome data, and report covariate-stratified τ(x) coverage across covariate quintiles to substantiate calibration for heterogeneous effects beyond scalar summaries. We draw a clean distinction between tails-as-contamination (handled by Welsch + Huber nuisance) and tails-as-signal (handled by a tail-aware CATE basis); an empirical probe confirms a tail-aware basis recovers τtail with full subgroup coverage, while the library's Hill-estimator path is contamination-directed and should not be used for heterogeneous τ. We map six empirical boundaries (contamination ceiling, clean-data efficiency cost, basis sensitivity, sample size, treatment type, compute) and show where other tools are preferable. Code and reproducible benchmarks are released.

We adopt a residual network (ResNet) [23] based feature extractor, with ELU as the activation function. Following [15], we adopt group normalization and instance normalization for better stability of the networks. We adopt the "leave-one-out" training strategy for obtaining the results on each of the categories of MVTec-AD. All experiments are performed with the same settings and hyperparameters. We resize all images to 128 128, and do not perform any data augmentation.