We consider the problem of joint estimation of the parameters of $m$ linear dynamical systems, given access to single realizations of their respective trajectories, each of length $T$. The linear systems are assumed to reside on the nodes of an undirected and connected graph $G = ([m], \mathcal{E})$, and the system matrices are assumed to either vary smoothly or exhibit small number of ``jumps'' across the edges. We consider a total variation penalized least-squares estimator and derive non-asymptotic bounds on the mean squared error (MSE) which hold with high probability. In particular, the bounds imply for certain choices of well connected $G$ that the MSE goes to zero as $m$ increases, even when $T$ is constant. The theoretical results are supported by extensive experiments on synthetic and real data.

The identification of ordinary differential equations (ODEs) and dynamical systems is a fundamental problem in control [32, 59, 60], data assimilation [42, 84], and more recently in scientific machine learning (ML) [11, 72, 74]. While algorithms such as Sparse Identification of Nonlinear Dynamics (SINDy) and its variants [46] are widely used by practitioners, they often fail in scenarios where observations of the state of the system are scarce, indirect, and noisy. In such scenarios modifications to SINDy-type methods are required to enforce additional constraints on the recovered equations to make them consistent with the observational data. Put simply, traditional SINDy-type methods work in two steps: (1) the data is used to filter the state of the system and estimate the derivatives, and (2) the filtered state is used to learn the underlying dynamics. In the regime of scarce, noisy and incomplete data, step 1 is inaccurate, which can propagate to poor results in the subsequent step 2. In this paper, we propose an all-at-once approach to filtering and equation learning based on collocation in a reproducing kernel Hilbert space (RKHS) which we term Joint SINDy (JSINDy), and shows that the issues above can be mitigated by performing both steps together. This joins a broader class of dynamics-informed methods that integrate the governing equations directly into the learning objective, either as hard constraints or as least-squares relaxations, which couples the problems of state estimation and model discovery. Representative examples include physics-informed and sparse-regression frameworks based on neural networks, splines, kernels, finite differences, and adjoint methods [21, 27, 39, 41, 72, 73, 88].

We study the problem of matching correlated VAR time series databases, where a multivariate time series is observed along with a perturbed and permuted version, and the goal is to recover the unknown matching between them. To model this, we introduce a probabilistic framework in which two time series $(x_t)_{t\in[T]},(x^\#_t)_{t\in[T]}$ are jointly generated, such that $x^\#_t=x_{π^*(t)}+σ\tilde{x}_{π^*(t)}$, where $(x_t)_{t\in[T]},(\tilde{x}_t)_{t\in[T]}$ are independent and identically distributed vector autoregressive (VAR) time series of order $1$ with Gaussian increments, for a hidden $π^*$. The objective is to recover $π^*$, from the observation of $(x_t)_{t\in[T]},(x^\#_t)_{t\in[T]}$. This generalizes the classical problem of matching independent point clouds to the time series setting. We derive the maximum likelihood estimator (MLE), leading to a quadratic optimization over permutations, and theoretically analyze an estimator based on linear assignment. For the latter approach, we establish recovery guarantees, identifying thresholds for $σ$ that allow for perfect or partial recovery. Additionally, we propose solving the MLE by considering convex relaxations of the set of permutation matrices (e.g., over the Birkhoff polytope). This allows for efficient estimation of $π^*$ and the VAR parameters via alternating minimization. Empirically, we find that linear assignment often matches or outperforms MLE relaxation based approaches.

We propose a way of transforming the problem of conditional density estimation into a single nonparametric regression task via the introduction of auxiliary samples. This allows leveraging regression methods that work well in high dimensions, such as neural networks and decision trees. Our main theoretical result characterizes and establishes the convergence of our estimator to the true conditional density in the data limit. We develop condensité, a method that implements this approach. We demonstrate the benefit of the auxiliary samples on synthetic data and showcase that condensité can achieve good out-of-the-box results. We evaluate our method on a large population survey dataset and on a satellite imaging dataset. In both cases, we find that condensité matches or outperforms the state of the art and yields conditional densities in line with established findings in the literature on each dataset. Our contribution opens up new possibilities for regression-based conditional density estimation and the empirical results indicate strong promise for applied research.

Ecological Momentary Assessment provides real-time data on suicidal thoughts and behaviors, but predicting suicide attempts remains challenging due to their rarity and patient heterogeneity. We show that single models fit to all patients perform poorly, while individualized models improve performance but still overfit to patients with limited data. To address this, we introduce Latent Similarity Gaussian Processes (LSGPs) to capture patient heterogeneity, enabling those with little data to leverage similar patients' trends. Preliminary results show promise: even without kernel-design, we outperform all but one baseline while offering a new understanding of patient similarity.

In this work, we propose a set of conformal prediction procedures tailored to compositional responses, where outcomes are proportions that must be positive and sum to one. Building on Dirichlet regression, we introduce a split conformal approach based on quantile residuals and a highest-density region strategy that combines a fast coordinate-floor approximation with an internal grid refinement to restore sharpness. Both constructions are model-agnostic at the conformal layer and guarantee finite-sample marginal coverage under exchangeability, while respecting the geometry of the simplex. A comprehensive Monte Carlo study spanning homoscedastic and heteroscedastic designs shows that the quantile residual and grid-refined HDR methods achieve empirical coverage close to the nominal 90\% level and produce substantially narrower regions than the coordinate-floor approximation, which tends to be conservative. We further demonstrate the methods on household budget shares from the BudgetItaly dataset, using standardized socioeconomic and price covariates with a train, calibration, and test split. In this application, the grid-refined HDR attains coverage closest to the target with the smallest average widths, closely followed by the quantile residual approach, while the simple triangular HDR yields wider, less informative sets. Overall, the results indicate that conformal prediction on the simplex can be both calibrated and efficient, providing practical uncertainty quantification for compositional prediction tasks.

Accurately solving partial differential equations (PDEs) is critical to understanding complex scientific and engineering phenomena, yet traditional numerical solvers are computationally expensive. Surrogate models offer a more efficient alternative, but their development is hindered by the cost of generating sufficient training data from numerical solvers. In this paper, we present a novel framework for active learning (AL) in PDE surrogate modeling that reduces this cost. Unlike the existing AL methods for PDEs that always acquire entire PDE trajectories, our approach strategically generates only the most important time steps with the numerical solver, while employing the surrogate model to approximate the remaining steps. This dramatically reduces the cost incurred by each trajectory and thus allows the active learning algorithm to try out a more diverse set of trajectories given the same budget. To accommodate this novel framework, we develop an acquisition function that estimates the utility of a set of time steps by approximating its resulting variance reduction. We demonstrate the effectiveness of our method on several benchmark PDEs, including the Burgers' equation, Korteweg-De Vries equation, Kuramoto-Sivashinsky equation, the incompressible Navier-Stokes equation, and the compressible Navier-Stokes equation. Experiments show that our approach improves performance by large margins over the best existing method. Our method not only reduces average error but also the 99\%, 95\%, and 50\% quantiles of error, which is rare for an AL algorithm. All in all, our approach offers a data-efficient solution to surrogate modeling for PDEs.

This paper presents a real time, data driven decision support framework for epidemic control. We combine a compartmental epidemic model with sequential Bayesian inference and reinforcement learning (RL) controllers that adaptively choose intervention levels to balance disease burden, such as intensive care unit (ICU) load, against socio economic costs. We construct a context specific cost function using empirical experiments and expert feedback. We study two RL policies: an ICU threshold rule computed via Monte Carlo grid search, and a policy based on a posterior averaged Q learning agent. We validate the framework by fitting the epidemic model to publicly available ICU occupancy data from the COVID 19 pandemic in England and then generating counterfactual roll out scenarios under each RL controller, which allows us to compare the RL policies to the historical government strategy. Over a 300 day period and for a range of cost parameters, both controllers substantially reduce ICU burden relative to the observed interventions, illustrating how Bayesian sequential learning combined with RL can support the design of epidemic control policies.

We study differentially private model training with stochastic gradient descent under learning rate scheduling and correlated noise. Although correlated noise, in particular via matrix factorizations, has been shown to improve accuracy, prior theoretical work focused primarily on the prefix-sum workload. That workload assumes a constant learning rate, whereas in practice learning rate schedules are widely used to accelerate training and improve convergence. We close this gap by deriving general upper and lower bounds for a broad class of learning rate schedules in both single- and multi-epoch settings. Building on these results, we propose a learning-rate-aware factorization that achieves improvements over prefix-sum factorizations under both MaxSE and MeanSE error metrics. Our theoretical analysis yields memory-efficient constructions suitable for practical deployment, and experiments on CIFAR-10 and IMDB datasets confirm that schedule-aware factorizations improve accuracy in private training.

Intelligent agents equipped with causal knowledge can optimize their action spaces to avoid unnecessary exploration. The structural causal bandit framework provides a graphical characterization for identifying actions that are unable to maximize rewards by leveraging prior knowledge of the underlying causal structure. While such knowledge enables an agent to estimate the expected rewards of certain actions based on others in online interactions, there has been little guidance on how to transfer information inferred from arbitrary combinations of datasets collected under different conditions -- observational or experimental -- and from heterogeneous environments. In this paper, we investigate the structural causal bandit with transportability, where priors from the source environments are fused to enhance learning in the deployment setting. We demonstrate that it is possible to exploit invariances across environments to consistently improve learning. The resulting bandit algorithm achieves a sub-linear regret bound with an explicit dependence on informativeness of prior data, and it may outperform standard bandit approaches that rely solely on online learning.