Conformal prediction is often calibrated with a single pooled threshold, but this can hide cross-group heterogeneity in score distributions and distort group-wise coverage. We study this phenomenon through the population score distributions underlying split conformal calibration. First, we derive a conservation law and lower bound showing that pooled calibration incurs irreducible group-wise coverage distortion at a scale set by cross-group quantile heterogeneity. Second, we demonstrate that the two leading fairness definitions for conformal prediction, Equalized Coverage and Equalized Set Size, are fundamentally in tension. Third, we quantify the cost of moving between policies which treat groups separately or pool them. Experiments on synthetic and real data confirm the same bidirectional trade-off after finite-sample calibration. Our results show that, for the policy families studied here, calibration choice does not remove cross-group heterogeneity; it determines whether the resulting distortion appears in the coverage or size dimension, providing a principled lens for analyzing fairness-oriented calibration choices in practice.

We propose a data augmentation method for offline reinforcement learning, motivated by active positioning problems. Particularly, our approach enables the training of off-policy models from a limited number of suboptimal trajectories. We introduce a trajectory-based augmentation technique that exploits task structure and the geometric relationship between rewards, value functions, and mathematical properties of logging policies. During data collection, our augmentation supports suboptimal logging policies, leading to higher data quality and improved offline reinforcement learning performance. We provide theoretical justification for these strategies and validate them empirically across positioning tasks of varying dimensionality and under partial observability.

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.

Spectral graph sparsification is a classical tool for reducing graph complexity while preserving Laplacian quadratic forms. In graph neural networks (GNNs), sparsification is often used to accelerate computation while maintaining predictive performance. In this work, we study a complementary representation-level question: does sparsification preserve the geometry of learned embeddings? For polynomial-filter GNNs, we prove that any $ε$-spectral sparsifier induces $O(ε)$ perturbations in polynomial graph filters, multilayer hidden representations, and their Gram matrices. These guarantees imply stability of squared pairwise distances, class means, and covariance structure in embedding space. We further establish finite-time training stability: under smoothness and boundedness assumptions, gradient descent on dense and sparsified graphs produces weight trajectories whose separation grows at most proportionally to the sparsification distortion. Empirically, effective-resistance sparsification validates the predicted perturbation chain on synthetic graphs and preserves hidden representation geometry on real datasets. In our experiments, the gram matrix and training dynamics show low divergence even under substantial sparsification, consistent with the predicted stability under spectral sparsification. Hidden Gram preservation strongly predicts neighborhood preservation and class-centroid stability across FashionMNIST, Cora, and Paul15. Together, these results show that spectral sparsification preserves not only graph operators, but also the representation geometry that supports downstream use of GNN embeddings for interpretability.

We only observe their transformed versions h(xis) and g(xit), for some known function class h() and g(). Our goal is to perform a statistical test checking if Psource = Ptarget while removing the distortions induced by the transformations. This problem is closely related to domain adaptation, and in our case, is motivated by the need to combine clinical and imaging based biomarkers from multiple sites and/or batches - a fairly common impediment in conducting analyses with much larger sample sizes. We address this problem using ideas from hypothesis testing on the transformed measurements, wherein the distortions need to be estimated in tandem with the testing. We derive a simple algorithm and study its convergence and consistency properties in detail, and provide lower-bound strategies based on recent work in continuous optimization. On a dataset of individuals at risk for Alzheimer's disease, our framework is competitive with alternative procedures that are twice as expensive and in some cases operationally infeasible to implement.

A.1 Training dataset pre-processing We used 40000publicly available videos from YouTube which were available in a spatial resolution of at least 1920 1080 pixels. In an attempt not to skew the distribution of content too far from what may inform biological representation learning, we excluded most artificial content such as screenshots and videos of computer games. To reduce video compression artifacts and prevent systematic downsampling artifacts, each segment was then spatially downsampled to a randomized height between 128 and 160. Each segment was then separated into 15 pairs of neighboring frames, and a randomly placed, but spatially colocated patch of 64 64 pixels was cropped out of each frame pair. The order of the frame pairs was then randomized in a running buffer, and all RGB pixel values were normalized to the range between 0 and 1 before being fed into the model.

The details of multiple datasets for OIQA task are presented in Table A. For the dataset that contains scanpath coordinates, we can directly sample viewport sequences from it and use our network to predict the quality scores. However, it is challenging and costly to record user scanpath data for every ODI in realistic scenarios. The scanpath information is likely unavailable when evaluating the quality of a panorama. Therefore, we propose a generalized Recursive Probability Sampling (RPS) method to generate multiple pseudo viewport sequences for the panorama, which assists the network to predict an accurate quality score in a way that is similar to the observer's actual scoring process. In JUFE and JXUFE, each ODI consists of 300 viewport coordinates, recorded using a head-mounted display (HMD).

We study a new family of inverse problems for recovering representations of corrupted data. We assume access to a pre-trained representation learning network R(x) that operates on clean images, like CLIP. The problem is to recover the representation of an image R(x), if we are only given a corrupted version A(x), for some known forward operator A. We propose a supervised inversion method that uses a contrastive objective to obtain excellent representations for highly corrupted images. Using a linear probe on our robust representations, we achieve a higher accuracy than end-to-end supervised baselines when classifying images with various types of distortions, including blurring, additive noise, and random pixel masking. We evaluate on a subset of ImageNet and observe that our method is robust to varying levels of distortion. Our method outperforms end-to-end baselines even with a fraction of the labeled data in a wide range of forward operators.