In many longitudinal studies, a large number of variables are measured repeatedly over time, with substantial missing data. Existing methods, such as probabilistic principal component analysis (PPCA), are ill-equipped to handle such incomplete, high-dimensional longitudinal data, as they fail to account for the nested sources of variation and temporal dependency inherent in repeated measures. We introduce hierarchical probabilistic principal component analysis (HPPCA), a two-level probabilistic factor model that explicitly separates between-subject variance from time-varying within-subject dynamics. The within-subject latent factors are modeled by a Gaussian process. We develop an EM algorithm to handle missing data and flexible covariance kernels, accelerated by computationally efficient initializers. Simulation studies demonstrated that HPPCA robustly recovers model parameters subspaces and substantially outperforms both standard PPCA and multivariate functional PCA in imputation accuracy, even under heavy missingness and model misspecification. An application to the long COVID symptoms in the Researching COVID to Enhance Recovery adult cohort revealed that HPPCA effectively captured the data's hierarchical structure and its learned features significantly improved the prediction of clinical outcomes and the recovery of masked clinical records compared to exisiting methods.

A central question in computational neuroscience is how structure determines functioninneuralnetworks.

A.2 Proof of Relation (3) We can write D One class of transport maps we consider in our numerical experiments (i.e., to approximate Another underlying class of transports that we use in our numerical experiments are inverse auto-regressive flows (IAFs). IAFs are built as a composition of component-wise affine transformations, where the shift and scaling functions of each component only depend on earlier indexed variables. Flows are typically comprised of several IAF stages with the components either randomly permuted or, as we choose, reversed in between each stage. Here we discuss how generalized linear models may naturally admit lazy structure. Here we describe the numerical algorithms required by the lazy map framework.

See Singh et al. (2000). Moreover, SARSA and Expected SARSA are also both appropriate, if the agent is greedy in the limit. Note that condition 2 requires that the agent takes every action in every state infinitely many times Proof. Let A satisfy the following in a given NDP: A is greedy in the limit, i.e. for all δ > 0, P (Q A's Q-values are accurate in the limit, i.e. if π Then φ has a fixed point. Theorem 3. Every continuous NDP has a strongly ratifiable policy.

Generative artificial intelligence (AI) is rapidly transforming medical imaging by enabling capabilities such as data synthesis, image enhancement, modality translation, and spatiotemporal modeling. This review presents a comprehensive and forward-looking synthesis of recent advances in generative modeling including generative adversarial networks (GANs), variational autoencoders (VAEs), diffusion models, and emerging multimodal foundation architectures and evaluates their expanding roles across the clinical imaging continuum. We systematically examine how generative AI contributes to key stages of the imaging workflow, from acquisition and reconstruction to cross-modality synthesis, diagnostic support, and treatment planning. Emphasis is placed on both retrospective and prospective clinical scenarios, where generative models help address longstanding challenges such as data scarcity, standardization, and integration across modalities. To promote rigorous benchmarking and translational readiness, we propose a three-tiered evaluation framework encompassing pixel-level fidelity, feature-level realism, and task-level clinical relevance. We also identify critical obstacles to real-world deployment, including generalization under domain shift, hallucination risk, data privacy concerns, and regulatory hurdles. Finally, we explore the convergence of generative AI with large-scale foundation models, highlighting how this synergy may enable the next generation of scalable, reliable, and clinically integrated imaging systems. By charting technical progress and translational pathways, this review aims to guide future research and foster interdisciplinary collaboration at the intersection of AI, medicine, and biomedical engineering.