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.

Constructing minimum-volume prediction regions that satisfy conditional coverage is a fundamental challenge in multivariate regression. Standard approaches rely on explicitly estimating the full conditional density and subsequently thresholding it. This two-step plug-in process is notoriously difficult, sensitive to estimation errors, and computationally expensive. One would like to instead optimize the region directly. Formulating a direct solution is challenging, however, because it requires minimizing a volume objective that is coupled with the conditional quantiles of the model's own estimation error. In this work, we address this challenge. We introduce super-level-set regression (SLS), a novel mathematical framework that successfully resolves this implicit coupling, allowing us to directly parameterize and optimize the geometric boundaries of the target conditional level sets. By bypassing full distribution estimation and leveraging flexible volume-preserving frontier functions, our approach natively captures complex, multimodal, and disjoint conditional structures end-to-end. Ultimately, SLS offers a new perspective on multivariate conditional quantile regression, replacing the restrictive assumptions of density-first methods with a direct geometric optimization strategy.

Quantile regression is a fundamental tool for distributional learning but poses significant optimization challenges for deep models due to the non-smoothness of the pinball loss. We propose ConquerNet, a class of \textbf{con}volution-smoothed \textbf{qu}antil\textbf{e} \textbf{R}eLU neural \textbf{net}works, which yield smooth objectives while preserving the underlying quantile structure. We establish general nonasymptotic risk bounds for ConquerNet under mild conditions, providing minimax guarantees over Besov function classes. In numerical studies, we demonstrate that the proposed approach outperforms standard quantile neural networks at multiple quantile levels, showing improved estimation accuracy and training efficiency across the board, with particularly pronounced advantages at high and low quantiles.

When modeling class-imbalanced data, it is crucial to address the imbalance, as models trained on such data tend to be biased towards the majority classes. This problem is amplified under partial supervision, where pseudo-labels for unlabeled data are predicted based on imbalanced labeled data, propagating the bias. While recent semi-supervised models address class imbalance, they typically assume single-modal input data. However, with the growing availability of multimodal data, it is essential to leverage complementary modalities. In this article, we propose a multimodal deep generative model for semi-supervised learning under class imbalance. Our approach uses separate encoders for each modality, sharing latent variables across modalities, and simplifies joint posterior computation with a product-of-experts method. To further address class imbalance, we replace typical Gaussian distributions with Student's t-distributions for the prior, encoder, and decoder, better capturing the heavy-tailed latent distributions in imbalanced data. We derive a new objective function for training the proposed model on both labeled and unlabeled data using $γ$-power divergence. Empirical results on benchmark and real-world datasets demonstrate that our model outperforms baseline methods in generalization, achieving superior classification performance for partially labeled multimodal data with imbalanced class distributions.

We introduce the metagame, a conceptual framework for quantifying second-order interaction effects of model explanations. For any first-order attribution $ϕ(f)$ explaining a model $f$, we measure the directional influence of feature $j$ on the attribution of feature $i$, denoted as meta-attribution $φ_{j \to i}(f)$, by treating the attribution method itself as a cooperative game and computing its Shapley value. Theoretically, we prove that attributions hierarchically decompose into meta-attributions, and establish these as directional extensions of existing interaction indices. Empirically, we demonstrate that the metagame delivers insights across diverse interpretability applications: (i) quantifying token interactions in instruction-tuned language models, (ii) explaining cross-modal similarity in vision-language encoders, and (iii) interpreting text-to-image concepts in multimodal diffusion transformers.

Learning identifiable representations in deep generative models remains a fundamental challenge, particularly for sequential data with regime-switching dynamics. Existing approaches establish identifiability under restrictive assumptions, such as stationarity or limited emission models, and typically rely on variational autoencoder (VAE) estimators, which introduce approximation gaps that limit the recovery of the latent structure. In this work, we address both the theoretical and practical limitations of this setting. First, we establish identifiability of a broad class of recurrent nonlinear switching dynamical systems under flexible assumptions, significantly extending prior results. Second, we introduce $Ω$SDS, a flow-based estimator that enables exact likelihood optimization using expectation-maximisation. Through empirical validation on both synthetic and real-world data, our results demonstrate that $Ω$SDS achieves improved disentanglement compared to VAE-based estimators and more accurate forecasting of underlying dynamics.

Cocoa (Theobroma cacao) is a critical cash crop for millions of smallholder farmers in West Africa, where Cocoa Swollen Shoot Virus Disease (CSSVD) and anthracnose cause devastating yield losses. Automated disease detection from leaf images is essential for early intervention, yet deploying such systems in resource-constrained settings demands models that are small, fast, and require no internet connectivity. Existing edge-deployable plant disease systems rely on end-to-end deep learning without uncertainty quantification, while Bayesian methods for edge devices focus on hardware-level inference architectures rather than agricultural applications. We bridge this gap with TinyBayes, the first framework to combine a closed-form Bayesian classifier with a mobile-grade computer vision pipeline for crop disease detection. Our pipeline uses YOLOv8-Nano (5.9 MB) for lesion localisation, MobileNetV3-Small (3.5 MB) for feature extraction, and the Jacobi prior; a Bayesian method that provides a closed form non-iterative estimators via projection, for the classification. The Jacobi-DMR (Distributed Multinomial Regression) classifier adds only 13.5 KB to the pipeline, bringing the total model size within 9.5 MB, while achieving 78.7% accuracy on the Amini Cocoa Contamination Challenge dataset and enabling end-to-end CPU inference under 150 ms per image. We benchmark against seven classifiers including Random Forest, SVM, Ridge, Lasso, Elastic Net, XGBoost, and Jacobi-GP, and demonstrate that the Jacobi-DMR offers the best trade-off between accuracy, model size, and inference speed for edge deployment. We have proved the asymptotic equivalence and consistency, asymptotic normality and the bias correction of Jacobi-DMR. All data and codes are available here: https://github.com/shouvik-sardar/TinyBayes

We study the grokking phenomenon through the lens of topology. Using persistent homology on point clouds derived from the embedding matrices of a range of models trained on modular arithmetic with varying primes, we identify a clear and consistent topological signature of grokking: a sharp increase in both the maximum and total persistence of first homology ($H_1$). Persistence diagrams reveal the emergence of a dominant long-lived topological feature together with increasingly structured secondary features, reflecting the underlying cyclic structure of the task. Compared to existing spectral and geometric diagnostics -- specifically, Fourier analysis and local intrinsic dimension -- persistent homology provides a unified geometric and topological characterization of representation learning, capturing both local and global multi-scale structure. Ablations across data regimes and control settings show that these topological transitions are tied to generalization rather than memorization. Our results suggest that persistent homology offers a principled and interpretable framework for analyzing how neural networks internalize latent structure during training.

Order-Agnostic autoregressive models have demonstrated strong performance in deep generative modeling, yet their use in settings with incomplete data remains largely unexplored. In this work, we reinterpret them through the lens of missing data. First, we show that their standard training procedure on fully observed data implicitly performs imputation under a missing completely at random mechanism, resulting in robust out-of-sample imputation performance in settings with high missingness. Second, we introduce the first principled framework for training them directly on incomplete datasets under general missingness mechanisms. Third, we leverage their amortized conditional density estimation to perform active information acquisition, i.e., sequentially selecting the most informative missing variables for downstream prediction or inference. Across a suite of real-world benchmarks, our Missingness-Aware Order-Agnostic Autoregressive Model (MO-ARM) consistently outperforms established imputation baselines.

Real-world datasets are inherently heterogeneous, yet how per-class structural differences and sampling imbalance shape the training dynamics of diffusion models-and potentially exacerbate disparities-remains poorly understood. While models typically transition from an initial phase of generalization to memorizing the training set, existing theory assumes homogeneous data, leaving open how class imbalance and heterogeneity reshape these dynamics. In this work, we develop a high-dimensional analytical framework to study class-dependent learning in score-based diffusion models. Analyzing a random-features model trained on Gaussian mixtures, we derive the feature-covariance spectrum to characterize per-class generalization and memorization times. We reveal the explicit hierarchy governing these dynamics: class variance is the primary determinant of learning order-consistently favoring higher-variance classes-while centroid geometry plays a secondary role. Sampling imbalance acts as a modulator that can reverse this ordering and, under strong imbalance, forces minority classes to acquire distinct, delayed speciation times during backward diffusion. Together, these results suggest that diffusion models can memorize some classes while others remain insufficiently learned. We validate our theoretical predictions empirically using U-Net models trained on Fashion MNIST.