A common approach to perform PCA on probability measures is to embed them into a Hilbert space where standard functional PCA techniques apply. While convergence rates for estimating the embedding of a single measure from $m$ samples are well understood, the literature has not addressed the setting involving multiple measures. In this paper, we study PCA in a double asymptotic regime where $n$ probability measures are observed, each through $m$ samples. We derive convergence rates of the form $n^{-1/2} + m^{-α}$ for the empirical covariance operator and the PCA excess risk, where $α>0$ depends on the chosen embedding. This characterizes the relationship between the number $n$ of measures and the number $m$ of samples per measure, revealing a sparse (small $m$) to dense (large $m$) transition in the convergence behavior. Moreover, we prove that the dense-regime rate is minimax optimal for the empirical covariance error. Our numerical experiments validate these theoretical rates and demonstrate that appropriate subsampling preserves PCA accuracy while reducing computational cost.

A/B testing has become a gold standard for modern technological companies to conduct policy evaluation. Yet, its application to time series experiments, where policies are sequentially assigned over time, remains challenging. Existing designs suffer from two limitations: (i) they do not fully leverage the entire history for treatment allocation; (ii) they rely on strong assumptions to approximate the objective function (e.g., the mean squared error of the estimated treatment effect) for optimizing the design. We first establish an impossibility theorem showing that failure to condition on the full history leads to suboptimal designs, due to the dynamic dependencies in time series experiments. To address both limitations simultaneously, we next propose a transformer reinforcement learning (RL) approach which leverages transformers to condition allocation on the entire history and employs RL to directly optimize the MSE without relying on restrictive assumptions. Empirical evaluations on synthetic data, a publicly available dispatch simulator, and a real-world ridesharing dataset demonstrate that our proposal consistently outperforms existing designs.

Conformal Prediction (CP) provides a statistical framework for uncertainty quantification that constructs prediction sets with coverage guarantees. While CP yields uncontrolled prediction set sizes, Backward Conformal Prediction (BCP) inverts this paradigm by enforcing a predefined upper bound on set size and estimating the resulting coverage guarantee. However, the looseness induced by Markov's inequality within the BCP framework causes a significant gap between the estimated coverage bound and the empirical coverage. In this work, we introduce ST-BCP, a novel method that introduces a data-dependent transformation of nonconformity scores to narrow the coverage gap. In particular, we develop a computable transformation and prove that it outperforms the baseline identity transformation. Extensive experiments demonstrate the effectiveness of our method, reducing the average coverage gap from 4.20\% to 1.12\% on common benchmarks.

Data-driven algorithm design automates hyperparameter tuning, but its statistical foundations remain limited because model performance can depend on hyperparameters in implicit and highly non-smooth ways. Existing guarantees focus on the simple case of a one-dimensional (scalar) hyperparameter. This leaves the practically important, multi-dimensional hyperparameter tuning setting unresolved. We address this open question by establishing the first general framework for establishing generalization guarantees for tuning multi-dimensional hyperparameters in data-driven settings. Our approach strengthens the generalization guarantee framework for semi-algebraic function classes by exploiting tools from real algebraic geometry, yielding sharper, more broadly applicable guarantees. We then extend the analysis to hyperparameter tuning using the validation loss under minimal assumptions, and derive improved bounds when additional structure is available. Finally, we demonstrate the scope of the framework with new learnability results, including data-driven weighted group lasso and weighted fused lasso.

Abstract--We introduce SNAP (Self-coNsistent Agreement Principle), a self-supervised framework for robust computation based on mutual agreement. Based on an Agreement-Reliability Hypothesis SNAP assigns weights that quantify agreement, emphasizing trustworthy items and downweighting outliers without supervision or prior knowledge. A key result is the Exponential Suppression of Outlier W eights, ensuring that outliers contribute negligibly to computations, even in high-dimensional settings. We study properties of SNAP weighting scheme and show its practical benefits on vector averaging and subspace estimation. Particularly, we demonstrate that non-iterative SNAP outperforms the iterative Weiszfeld algorithm and two variants of multivariate median of means. SNAP thus provides a flexible, easy-to-use, broadly applicable approach to robust computation. In many computational and machine learning tasks, multiple candidate solutions, model predictions, or observed entities are available. Some are reliable, while others are noisy or erroneous.

In biomedical research, repeated measurements within each subject are often processed to remove artifacts and unwanted sources of variation. The resulting data are used to construct derived outcomes that act as proxies for scientific outcomes that are not directly observable. Although intra-subject processing is widely used, its impact on inter-subject statistical inference has not been systematically studied, and a principled framework for causal analysis in this setting is lacking. In this article, we propose a semiparametric framework for causal inference with derived outcomes obtained after intra-subject processing. This framework applies to settings with a modular structure, where intra-subject analyses are conducted independently across subjects and are followed by inter-subject analyses based on parameters from the intra-subject stage. We develop multiply robust estimators of causal parameters under rate conditions on both intra-subject and inter-subject models, which allows the use of flexible machine learning. We specialize the framework to a mediation setting and focus on the natural direct effect. For high dimensional inference, we employ a step-down procedure that controls the exceedance rate of the false discovery proportion. Simulation studies demonstrate the superior performance of the proposed approach. We apply our method to estimate the impact of stimulant medication on brain connectivity in children with autism spectrum disorder.

Estimating causal effects from longitudinal trajectories is central to understanding the progression of complex conditions and optimizing clinical decision-making, such as comorbidities and long COVID recovery. We introduce \emph{C-kNN--LSH}, a nearest-neighbor framework for sequential causal inference designed to handle such high-dimensional, confounded situations. By utilizing locality-sensitive hashing, we efficiently identify ``clinical twins'' with similar covariate histories, enabling local estimation of conditional treatment effects across evolving disease states. To mitigate bias from irregular sampling and shifting patient recovery profiles, we integrate neighborhood estimator with a doubly-robust correction. Theoretical analysis guarantees our estimator is consistent and second-order robust to nuisance error. Evaluated on a real-world Long COVID cohort with 13,511 participants, \emph{C-kNN-LSH} demonstrates superior performance in capturing recovery heterogeneity and estimating policy values compared to existing baselines.

Flow-based methods have achieved significant success in various generative modeling tasks, capturing nuanced details within complex data distributions. However, few existing works have exploited this unique capability to resolve fine-grained structural details beyond generation tasks. This paper presents a flow-inspired framework for representation learning. First, we demonstrate that a rectified flow trained using independent coupling is zero everywhere at $t=0.5$ if and only if the source and target distributions are identical. We term this property the \emph{zero-flow criterion}. Second, we show that this criterion can certify conditional independence, thereby extracting \emph{sufficient information} from the data. Third, we translate this criterion into a tractable, simulation-free loss function that enables learning amortized Markov blankets in graphical models and latent representations in self-supervised learning tasks. Experiments on both simulated and real-world datasets demonstrate the effectiveness of our approach. The code reproducing our experiments can be found at: https://github.com/probabilityFLOW/zfe.

We introduce a transfer learning framework for regression that leverages heterogeneous source domains to improve predictive performance in a data-scarce target domain. Our approach learns a conditional generative model separately for each source domain and calibrates the generated responses to the target domain via conditional quantile matching. This distributional alignment step corrects general discrepancies between source and target domains without imposing restrictive assumptions such as covariate or label shift. The resulting framework provides a principled and flexible approach to high-quality data augmentation for downstream learning tasks in the target domain. From a theoretical perspective, we show that an empirical risk minimizer (ERM) trained on the augmented dataset achieves a tighter excess risk bound than the target-only ERM under mild conditions. In particular, we establish new convergence rates for the quantile matching estimator that governs the transfer bias-variance tradeoff. From a practical perspective, extensive simulations and real data applications demonstrate that the proposed method consistently improves prediction accuracy over target-only learning and competing transfer learning methods.

The ability of neural networks to learn useful features through stochastic gradient descent (SGD) is a cornerstone of their success. Most theoretical analyses focus on regression or on classification tasks with a positive margin, where worst-case gradient bounds suffice. In contrast, we study zero-margin nonlinear classification by analyzing the Gaussian XOR problem, where inputs are Gaussian and the XOR decision boundary determines labels. In this setting, a non-negligible fraction of data lies arbitrarily close to the boundary, breaking standard margin-based arguments. Building on Glasgow's (2024) analysis, we extend the study of training dynamics from discrete to Gaussian inputs and develop a framework for the dynamics of neuron blocks. We show that neurons cluster into four directions and that block-level signals evolve coherently, a phenomenon essential in the Gaussian setting where individual neuron signals vary significantly. Leveraging this block perspective, we analyze generalization without relying on margin assumptions, adopting an average-case view that distinguishes regions of reliable prediction from regions of persistent error. Numerical experiments confirm the predicted two-phase block dynamics and demonstrate their robustness beyond the Gaussian setting.