Single-cell RNA sequencing (scRNA-seq) has revolutionized our understanding of cellular processes by enabling gene expression analysis at the individual cell level. Clustering allows for the identification of cell types and the further discovery of intrinsic patterns in single-cell data. However, the high dimensionality and sparsity of scRNA-seq data continue to challenge existing clustering models. In this paper, we introduce JojoSCL, a novel self-supervised contrastive learning framework for scRNA-seq clustering. By incorporating a shrinkage estimator based on hierarchical Bayesian estimation, which adjusts gene expression estimates towards more reliable cluster centroids to reduce intra-cluster dispersion, and optimized using Stein's Unbiased Risk Estimate (SURE), JojoSCL refines both instance-level and cluster-level contrastive learning. Experiments on ten scRNA-seq datasets substantiate that JojoSCL consistently outperforms prevalent clustering methods, with further validation of its practicality through robustness analysis and ablation studies. JojoSCL's code is available at: https://github.com/ziwenwang28/JojoSCL.

Educators teaching entry-level university engineering modules face the challenge of identifying which topics students find most difficult and how to support diverse student needs effectively. This study demonstrates a rigorous yet interpretable statistical approach -- hierarchical Bayesian modeling -- that leverages detailed student response data to quantify both skill difficulty and individual student abilities. Using a large-scale dataset from an undergraduate Statics course, we identified clear patterns of skill mastery and uncovered distinct student subgroups based on their learning trajectories. Our analysis reveals that certain concepts consistently present challenges, requiring targeted instructional support, while others are readily mastered and may benefit from enrichment activities. Importantly, the hierarchical Bayesian method provides educators with intuitive, reliable metrics without sacrificing predictive accuracy. This approach allows for data-informed decisions, enabling personalized teaching strategies to improve student engagement and success. By combining robust statistical methods with clear interpretability, this study equips educators with actionable insights to better support diverse learner populations.

Reinforcement Learning (RL) has demonstrated tremendous empirical success across numerous challenging domains. However, we lack a strong theoretical understanding of the statistical complexity of RL in environments with large state spaces, where function approximation is required for sample-efficient learning. This thesis addresses this gap by rigorously examining the statistical complexity of RL with function approximation from a learning theoretic perspective. Departing from a long history of prior work, we consider the weakest form of function approximation, called agnostic policy learning, in which the learner seeks to find the best policy in a given class $Π$, with no guarantee that $Π$ contains an optimal policy for the underlying task. We systematically explore agnostic policy learning along three key axes: environment access -- how a learner collects data from the environment; coverage conditions -- intrinsic properties of the underlying MDP measuring the expansiveness of state-occupancy measures for policies in the class $Π$, and representational conditions -- structural assumptions on the class $Π$ itself. Within this comprehensive framework, we (1) design new learning algorithms with theoretical guarantees and (2) characterize fundamental performance bounds of any algorithm. Our results reveal significant statistical separations that highlight the power and limitations of agnostic policy learning.

Simulation-free training frameworks have been at the forefront of the generative modelling revolution in continuous spaces, leading to neural dynamical systems that encompass modern large-scale diffusion and flow matching models. Despite the scalability of training, the generation of high-quality samples and their corresponding likelihood under the model requires expensive numerical simulation -- inhibiting adoption in numerous scientific applications such as equilibrium sampling of molecular systems. In this paper, we revisit classical normalizing flows as one-step generative models with exact likelihoods and propose a novel, scalable training objective that does not require computing the expensive change of variable formula used in conventional maximum likelihood training. We propose Forward-Only Regression Training (FORT), a simple $\ell_2$-regression objective that maps prior samples under our flow to specifically chosen targets. We demonstrate that FORT supports a wide class of targets, such as optimal transport targets and targets from pre-trained continuous-time normalizing flows (CNF). We further demonstrate that by using CNF targets, our one-step flows allow for larger-scale training that exceeds the performance and stability of maximum likelihood training, while unlocking a broader class of architectures that were previously challenging to train. Empirically, we elucidate that our trained flows can perform equilibrium conformation sampling in Cartesian coordinates of alanine dipeptide, alanine tripeptide, and alanine tetrapeptide.

Score matching is a vital tool for learning the distribution of data with applications across many areas including diffusion processes, energy based modelling, and graphical model estimation. Despite all these applications, little work explores its use when data is incomplete. We address this by adapting score matching (and its major extensions) to work with missing data in a flexible setting where data can be partially missing over any subset of the coordinates. We provide two separate score matching variations for general use, an importance weighting (IW) approach, and a variational approach. We provide finite sample bounds for our IW approach in finite domain settings and show it to have especially strong performance in small sample lower dimensional cases. Complementing this, we show our variational approach to be strongest in more complex high-dimensional settings which we demonstrate on graphical model estimation tasks on both real and simulated data.

Varying coefficient models are popular for estimating nonlinear regression functions in functional data models. Their Bayesian variants have received limited attention in large data applications, primarily due to prohibitively slow posterior computations using Markov chain Monte Carlo (MCMC) algorithms. We introduce Bayesian data sketching for varying coefficient models to obviate computational challenges presented by large sample sizes. To address the challenges of analyzing large data, we compress the functional response vector and predictor matrix by a random linear transformation to achieve dimension reduction and conduct inference on the compressed data. Our approach distinguishes itself from several existing methods for analyzing large functional data in that it requires neither the development of new models or algorithms, nor any specialized computational hardware while delivering fully model-based Bayesian inference. Well-established methods and algorithms for varying coefficient regression models can be applied to the compressed data. We establish posterior contraction rates for estimating the varying coefficients and predicting the outcome at new locations with the randomly compressed data model. We use simulation experiments and analyze remote sensed vegetation data to empirically illustrate the inferential and computational efficiency of our approach.

Knowledge of accurate relative skills in any competitive system is essential, but foundational approaches such as ELO discard extremely relevant performance data by concentrating exclusively on binary outcomes. While margin of victory (MOV) extensions exist, they often lack a definitive method for incorporating this information. We introduce Margin of Victory Differential Analysis (MOVDA), a framework that enhances traditional rating systems by using the deviation between the true MOV and a $\textit{modeled expectation}$. MOVDA learns a domain-specific, non-linear function (a scaled hyperbolic tangent that captures saturation effects and home advantage) to predict expected MOV based on rating differentials. Crucially, the $\textit{difference}$ between the true and expected MOV provides a subtle and weighted signal for rating updates, highlighting informative deviations in all levels of contests. Extensive experiments on professional NBA basketball data (from 2013 to 2023, with 13,619 games) show that MOVDA significantly outperforms standard ELO and Bayesian baselines. MOVDA reduces Brier score prediction error by $1.54\%$ compared to TrueSkill, increases outcome accuracy by $0.58\%$, and most importantly accelerates rating convergence by $13.5\%$, while maintaining the computational efficiency of the original ELO updates. MOVDA offers a theoretically motivated, empirically superior, and computationally lean approach to integrating performance magnitude into skill rating for competitive environments like the NBA.

This paper explores the challenges and benefits of a trainable destruction process in diffusion samplers -- diffusion-based generative models trained to sample an unnormalised density without access to data samples. Contrary to the majority of work that views diffusion samplers as approximations to an underlying continuous-time model, we view diffusion models as discrete-time policies trained to produce samples in very few generation steps. We propose to trade some of the elegance of the underlying theory for flexibility in the definition of the generative and destruction policies. In particular, we decouple the generation and destruction variances, enabling both transition kernels to be learned as unconstrained Gaussian densities. We show that, when the number of steps is limited, training both generation and destruction processes results in faster convergence and improved sampling quality on various benchmarks. Through a robust ablation study, we investigate the design choices necessary to facilitate stable training. Finally, we show the scalability of our approach through experiments on GAN latent space sampling for conditional image generation.

Data-carving methods perform selective inference by conditioning the distribution of data on the observed selection event. However, existing data-carving approaches typically require an analytically tractable characterization of the selection event. This paper introduces a new method that leverages tools from flow-based generative modeling to approximate a potentially complex conditional distribution, even when the underlying selection event lacks an analytical description -- take, for example, the data-adaptive tuning of model parameters. The key idea is to learn a transport map that pushes forward a simple reference distribution to the conditional distribution given selection. This map is efficiently learned via a normalizing flow, without imposing any further restrictions on the nature of the selection event. Through extensive numerical experiments on both simulated and real data, we demonstrate that this method enables flexible selective inference by providing: (i) valid p-values and confidence sets for adaptively selected hypotheses and parameters, (ii) a closed-form expression for the conditional density function, enabling likelihood-based and quantile-based inference, and (iii) adjustments for intractable selection steps that can be easily integrated with existing methods designed to account for the tractable steps in a selection procedure involving multiple steps.

Small Uncrewed Aerial Systems (sUAS) are increasingly deployed as autonomous swarms in search-and-rescue and other disaster-response scenarios. In these settings, they use computer vision (CV) to detect objects of interest and autonomously adapt their missions. However, traditional CV systems often struggle to recognize unfamiliar objects in open-world environments or to infer their relevance for mission planning. To address this, we incorporate large language models (LLMs) to reason about detected objects and their implications. While LLMs can offer valuable insights, they are also prone to hallucinations and may produce incorrect, misleading, or unsafe recommendations. To ensure safe and sensible decision-making under uncertainty, high-level decisions must be governed by cognitive guardrails. This article presents the design, simulation, and real-world integration of these guardrails for sUAS swarms in search-and-rescue missions.