We develop a semi-parametric state-space model for time-series data with latent regime transitions. Classical Markov-switching models use fixed parametric transition functions, such as logistic or probit links, which restrict flexibility when transitions depend on nonlinear and context-dependent effects. We replace this assumption with learned functions $f_0, f_1 \in \calH$, where $\calH$ is either a reproducing kernel Hilbert space or a spline approximation space, and define transition probabilities as $p_{jk,t} = \sigmoid(f(\bx_{t-1}))$. The transition functions are estimated jointly with emission parameters using a generalized Expectation-Maximization algorithm. The E-step uses the standard forward-backward recursion, while the M-step reduces to a penalized regression problem with weights from smoothed occupation measures. We establish identifiability conditions and provide a consistency argument for the resulting estimators. Experiments on synthetic data show improved recovery of nonlinear transition dynamics compared to parametric baselines. An empirical study on financial time series demonstrates improved regime classification and earlier detection of transition events.

Explicit density evaluation is required in many statistical procedures, while synthesis of novel examples can enable agents to imagine and plan in an environment prior tochoosing aaction.

We show that learning in the kernel regime yields smooth interpolants, minimizing curvature, and reduces tocubic splines for uniform initializations.

Although extensivework hasrecently been done inthisfield, robustly distilling explicit model forms from very sparse data with considerable noise remains intractable.

Boosting has garnered significant interest across both machine learning and statistical communities. Traditional boosting algorithms, designed for fully observed random samples, often struggle with real-world problems, particularly with interval-censored data. This type of data is common in survival analysis and time-to-event studies where exact event times are unobserved but fall within known intervals. Effective handling of such data is crucial in fields like medical research, reliability engineering, and social sciences. In this work, we introduce novel nonparametric boosting methods for regression and classification tasks with interval-censored data. Our approaches leverage censoring unbiased transformations to adjust loss functions and impute transformed responses while maintaining model accuracy. Implemented via functional gradient descent, these methods ensure scalability and adaptability. We rigorously establish their theoretical properties, including optimality and mean squared error trade-offs. Our proposed methods not only offer a robust framework for enhancing predictive accuracy in domains where interval-censored data are common but also complement existing work, expanding the applicability of existing boosting techniques. Empirical studies demonstrate robust performance across various finite-sample scenarios, highlighting the practical utility of our approaches.

Neural operator surrogates for time-dependent partial differential equations (PDEs) conventionally employ autoregressive prediction schemes, which accumulate error over long rollouts and require uniform temporal discretization. We introduce the Continuous Flow Operator (CFO), a framework that learns continuous-time PDE dynamics without the computational burden of standard continuous approaches, e.g., neural ODE. The key insight is repurposing flow matching to directly learn the right-hand side of PDEs without backpropagating through ODE solvers. CFO fits temporal splines to trajectory data, using finite-difference estimates of time derivatives at knots to construct probability paths whose velocities closely approximate the true PDE dynamics. A neural operator is then trained via flow matching to predict these analytic velocity fields. This approach is inherently time-resolution invariant: training accepts trajectories sampled on arbitrary, non-uniform time grids while inference queries solutions at any temporal resolution through ODE integration. Across four benchmarks (Lorenz, 1D Burgers, 2D diffusion-reaction, 2D shallow water), CFO demonstrates superior long-horizon stability and remarkable data efficiency. CFO trained on only 25% of irregularly subsampled time points outperforms autoregressive baselines trained on complete data, with relative error reductions up to 87%. Despite requiring numerical integration at inference, CFO achieves competitive efficiency, outperforming autoregressive baselines using only 50% of their function evaluations, while uniquely enabling reverse-time inference and arbitrary temporal querying.