This paper introduces a comprehensive unified framework for constructing multi-view diffusion geometries through intertwined multi-view diffusion trajectories (MDTs), a class of inhomogeneous diffusion processes that iteratively combine the random walk operators of multiple data views. Each MDT defines a trajectory-dependent diffusion operator with a clear probabilistic and geometric interpretation, capturing over time the interplay between data views. Our formulation encompasses existing multi-view diffusion models, while providing new degrees of freedom for view interaction and fusion. We establish theoretical properties under mild assumptions, including ergodicity of both the point-wise operator and the process in itself. We also derive MDT-based diffusion distances, and associated embeddings via singular value decompositions. Finally, we propose various strategies for learning MDT operators within the defined operator space, guided by internal quality measures. Beyond enabling flexible model design, MDTs also offer a neutral baseline for evaluating diffusion-based approaches through comparison with randomly selected MDTs. Experiments show the practical impact of the MDT operators in a manifold learning and data clustering context.

We consider the problem of generalization of arbitrarily overparameterized two-layer ReLU Neural Networks with univariate input. Recent work showed that under square loss, flat solutions (motivated by flat / stable minima and Edge of Stability phenomenon) provably cannot overfit, but it remains unclear whether the same phenomenon holds for logistic loss. This is a puzzling open problem because existing work on logistic loss shows that gradient descent with increasing step size converges to interpolating solutions (at infinity, for the margin-separable cases). In this paper, we prove that the \emph{flatness implied generalization} is more delicate under logistic loss. On the positive side, we show that flat solutions enjoy near-optimal generalization bounds within a region between the left-most and right-most \emph{uncertain} sets determined by each candidate solution. On the negative side, we show that there exist arbitrarily flat yet overfitting solutions at infinity that are (falsely) certain everywhere, thus certifying that flatness alone is insufficient for generalization in general. We demonstrate the effects predicted by our theory in a well-controlled simulation study.

Discrete event systems are present both in observations of nature, socio economical sciences, and industrial systems. Standard analysis approaches do not usually exploit their dual event / state nature: signals are either modeled as transition event sequences, emphasizing event order alignment, or as categorical or ordinal state timeseries, usually resampled a distorting and costly operation as the observation period and number of events grow. In this work we define state transition event timeseries (STE-ts) and propose a new Selective Temporal Hamming distance (STH) leveraging both transition time and duration-in-state, avoiding costly and distorting resampling on large databases. STH generalizes both resampled Hamming and Jaccard metrics with better precision and computation time, and an ability to focus on multiple states of interest. We validate these benefits on simulated and real-world datasets.

Diffusion-based solvers for partial differential equations (PDEs) are often bottle-necked by slow gradient-based test-time optimization routines that use PDE residuals for loss guidance. They additionally suffer from optimization instabilities and are unable to dynamically adapt their inference scheme in the presence of noisy PDE residuals. To address these limitations, we introduce PRISMA (PDE Residual Informed Spectral Modulation with Attention), a conditional diffusion neural operator that embeds PDE residuals directly into the model's architecture via attention mechanisms in the spectral domain, enabling gradient-descent free inference. We show that PRISMA has competitive accuracy, at substantially lower inference costs, compared to previous methods across five benchmark PDEs especially with noisy observations, while using 10x to 100x fewer denoising steps, leading to 15x to 250x faster inference. Given the ubiquitous presence of partial differential equations (PDEs) in almost every scientific discipline, there is a rapidly growing literature on using neural networks for solving PDEs (Raissi et al., 2019a; Lu et al., 2019). This includes seminal works in operator learning methods such as the Fourier Neural Operator (FNO) Li et al. (2020) that learns resolution-independent mappings between function spaces of input parameters a and solution fields u. However, a major limitation of these methods is their reliance on complete and clean observations of either a or u, a condition rarely met in real-world applications where data is inherently noisy and sparse. The rise of generative models has inspired another class of methods for solving PDEs by modeling the joint distribution of a and u using diffusion-based backbones (Huang et al., 2024; Y ao et al., 2025; Lim et al., 2023; Shu et al., 2023; Bastek et al., 2024; Jacobsen et al., 2025). These methods offer two key advantages over operator learning methods: (i) they generate full posterior distributions of a and/or u, enabling principled uncertainty quantification crucial for ill-posed inverse problems, and (ii) they naturally accommodate sparse observations during inference using likelihood-based and PDE residual-based loss guidance, termed diffusion posterior sampling or test-time optimization.

Many online learning algorithms, including classical online PCA methods, enforce explicit normalization steps that discard the evolving norm of the parameter vector. We show that this norm can in fact encode meaningful information about the underlying statistical structure of the problem, and that exploiting this information leads to improved learning behavior. Motivated by this principle, we introduce Implicitly Normalized Online PCA (INO-PCA), an online PCA algorithm that removes the unit-norm constraint and instead allows the parameter norm to evolve dynamically through a simple regularized update. We prove that in the high-dimensional limit the joint empirical distribution of the estimate and the true component converges to a deterministic measure-valued process governed by a nonlinear PDE. This analysis reveals that the parameter norm obeys a closed-form ODE coupled with the cosine similarity, forming an internal state variable that regulates learning rate, stability, and sensitivity to signal-to-noise ratio (SNR). The resulting dynamics uncover a three-way relationship between the norm, SNR, and optimal step size, and expose a sharp phase transition in steady-state performance. Both theoretically and experimentally, we show that INO-PCA consistently outperforms Oja's algorithm and adapts rapidly in non-stationary environments. Overall, our results demonstrate that relaxing norm constraints can be a principled and effective way to encode and exploit problem-relevant information in online learning algorithms.

Mean-field games (MFGs) study the Nash equilibrium of systems with a continuum of interacting agents, which can be formulated as the fixed-point of optimal control problems. They provide a unified framework for a variety of applications, including optimal transport (OT) and generative models. Despite their broad applicability, solving high-dimensional MFGs remains a significant challenge due to fundamental computational and analytical obstacles. In this work, we propose a particle-based deep Flow Matching (FM) method to tackle high-dimensional MFG computation. In each iteration of our proximal fixed-point scheme, particles are updated using first-order information, and a flow neural network is trained to match the velocity of the sample trajectories in a simulation-free manner. Theoretically, in the optimal control setting, we prove that our scheme converges to a stationary point sublinearly, and upgrade to linear (exponential) convergence under additional convexity assumptions. Our proof uses FM to induce an Eulerian coordinate (density-based) from a Lagrangian one (particle-based), and this also leads to certain equivalence results between the two formulations for MFGs when the Eulerian solution is sufficiently regular. Our method demonstrates promising performance on non-potential MFGs and high-dimensional OT problems cast as MFGs through a relaxed terminal-cost formulation.

Foundation models, and in particular large language models, can generate highly informative responses, prompting growing interest in using these ''synthetic'' outputs as data in empirical research and decision-making. This paper introduces the idea of a foundation prior, which shows that model-generated outputs are not as real observations, but draws from the foundation prior induced prior predictive distribution. As such synthetic data reflects both the model's learned patterns and the user's subjective priors, expectations, and biases. We model the subjectivity of the generative process by making explicit the dependence of synthetic outputs on the user's anticipated data distribution, the prompt-engineering process, and the trust placed in the foundation model. We derive the foundation prior as an exponential-tilted, generalized Bayesian update of the user's primitive prior, where a trust parameter governs the weight assigned to synthetic data. We then show how synthetic data and the associated foundation prior can be incorporated into standard statistical and econometric workflows, and discuss their use in applications such as refining complex models, informing latent constructs, guiding experimental design, and augmenting random-coefficient and partially linear specifications. By treating generative outputs as structured, explicitly subjective priors rather than as empirical observations, the framework offers a principled way to harness foundation models in empirical work while avoiding the conflation of synthetic ''facts'' with real data.

We introduce Smart Bayes, a new classification framework that bridges generative and discriminative modeling by integrating likelihood-ratio-based generative features into a logistic-regression-style discriminative classifier. From the generative perspective, Smart Bayes relaxes the fixed unit weights of Naive Bayes by allowing data-driven coefficients on density-ratio features. From a discriminative perspective, it constructs transformed inputs as marginal log-density ratios that explicitly quantify how much more likely each feature value is under one class than another, thereby providing predictors with stronger class separation than the raw covariates. To support this framework, we develop a spline-based estimator for univariate log-density ratios that is flexible, robust, and computationally efficient. Through extensive simulations and real-data studies, Smart Bayes often outperforms both logistic regression and Naive Bayes. Our results highlight the potential of hybrid approaches that exploit generative structure to enhance discriminative performance.

Abstract--Principal Component Analysis (PCA) and K-means constitute fundamental techniques in multivariate analysis. Although they are frequently applied independently or sequentially to cluster observations, the relationship between them, especially when K-means is used to cluster variables rather than observations, has been scarcely explored. This study seeks to address this gap by proposing an innovative method that analyzes the relationship between clusters of variables obtained by applying K-means on transposed data and the principal components of PCA. Our approach involves applying PCA to the original data and K-means to the transposed data set, where the original variables are converted into observations. The contribution of each variable cluster to each principal component is then quantified using measures based on variable loadings. This process provides a tool to explore and understand the clustering of variables and how such clusters contribute to the principal dimensions of variation identified by PCA. We analyze multiple data sets with varying variability structures (USArrests, Iris, Decathlon2) to show that the correspondence between clusters of variables and principal components depends on the data's inherent structure.

We study the multi-objective linear contextual bandit problem, where multiple possible conflicting objectives must be optimized simultaneously. We propose \texttt{MOL-TS}, the \textit{first} Thompson Sampling algorithm with Pareto regret guarantees for this problem. Unlike standard approaches that compute an empirical Pareto front each round, \texttt{MOL-TS} samples parameters across objectives and efficiently selects an arm from a novel \emph{effective Pareto front}, which accounts for repeated selections over time. Our analysis shows that \texttt{MOL-TS} achieves a worst-case Pareto regret bound of $\widetilde{O}(d^{3/2}\sqrt{T})$, where $d$ is the dimension of the feature vectors, $T$ is the total number of rounds, matching the best known order for randomized linear bandit algorithms for single objective. Empirical results confirm the benefits of our proposed approach, demonstrating improved regret minimization and strong multi-objective performance.