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.

Deployment of legged robots for navigating challenging terrains (e.g., stairs, slopes, and unstructured environments) has gained increasing preference over wheel-based platforms. In such scenarios, accurate odometry estimation is a preliminary requirement for stable locomotion, localization, and mapping. Traditional proprioceptive approaches, which rely on leg kinematics sensor modalities and inertial sensing, suffer from irrepressible vertical drift caused by frequent contact impacts, foot slippage, and vibrations, particularly affected by inaccurate roll and pitch estimation. Existing methods incorporate exteroceptive sensors such as LiDAR or cameras. Further enhancement has been introduced by leveraging gravity vector estimation to add additional observations on roll and pitch, thereby increasing the accuracy of vertical pose estimation. However, these approaches tend to degrade in feature-sparse or repetitive scenes and are prone to errors from double-integrated IMU acceleration. To address these challenges, we propose GaRLILEO, a novel gravity-aligned continuous-time radar-leg-inertial odometry framework. GaRLILEO decouples velocity from the IMU by building a continuous-time ego-velocity spline from SoC radar Doppler and leg kinematics information, enabling seamless sensor fusion which mitigates odometry distortion. In addition, GaRLILEO can reliably capture accurate gravity vectors leveraging a novel soft S2-constrained gravity factor, improving vertical pose accuracy without relying on LiDAR or cameras. Evaluated on a self-collected real-world dataset with diverse indoor-outdoor trajectories, GaRLILEO demonstrates state-of-the-art accuracy, particularly in vertical odometry estimation on stairs and slopes. We open-source both our dataset and algorithm to foster further research in legged robot odometry and SLAM. https://garlileo.github.io/GaRLILEO

Neural networks are parametric and powerful tools for function approximation, and the choice of architecture heavily influences their interpretability, efficiency, and generalization. In contrast, Gaussian processes (GPs) are nonparametric probabilistic models that define distributions over functions using a kernel to capture correlations among data points. However, these models become computationally expensive for large-scale problems, as they require inverting a large covariance matrix. Kolmogorov- Arnold networks (KANs), semi-parametric neural architectures, have emerged as a prominent approach for modeling complex functions with structured and efficient representations through spline layers. Kurkova Kolmogorov-Arnold networks (KKANs) extend this idea by reducing the number of spline layers in KAN and replacing them with Chebyshev layers and multi-layer perceptrons, thereby mapping inputs into higher-dimensional spaces before applying spline-based transformations. Compared to KANs, KKANs perform more stable convergence during training, making them a strong architecture for estimating operators and system modeling in dynamical systems. By enhancing the KKAN architecture, we develop a novel learning algorithm, distance-aware error for Kurkova-Kolmogorov networks (K-DAREK), for efficient and interpretable function approximation with uncertainty quantification. Our approach establishes robust error bounds that are distance-aware; this means they reflect the proximity of a test point to its nearest training points. Through case studies on a safe control task, we demonstrate that K-DAREK is about four times faster and ten times higher computationally efficiency than Ensemble of KANs, 8.6 times more scalable than GP by increasing the data size, and 50% safer than our previous work distance-aware error for Kolmogorov networks (DAREK).