ransac integration

To address the challenges of unfriendly interfaces, cumbersome configuration, and lack of GPU acceleration in large-scale semidefinite programming (SDP) solvers, this work designs and implements cuHALLaR—the first native Julia interface integrating a GPU-accelerated low-rank SDP solver. The interface uniformly supports both the standard SDPA format and the novel hybrid sparse–low-rank (HSLR) data format, offering a concise API for loading custom problem data, configuring solver parameters, and executing experiments end-to-end. Crucially, it incorporates HSLR structure directly into the solving framework, substantially improving scalability in both memory footprint and computational efficiency. Built-in examples—including matrix completion and maximum independent set SDP relaxations—demonstrate robustness and high performance on large-scale instances. Experimental results confirm that cuHALLaR delivers both numerical stability and speedup over CPU-based alternatives. This work provides an efficient, user-friendly, and ecosystem-native Julia solution for SDP research and practical applications.

To address the poor scalability of Gaussian process (GP)-driven parallel-in-time (PinT) methods—such as GParareal—in high-dimensional systems and large-scale parallel environments, this paper proposes nnGParareal. The key innovation is the first integration of nearest-neighbor sampling into the GP-PinT framework, combined with adaptive sample selection and neighbor-based data compression. This reduces the GP model complexity from O(N³) to O(N log N) while preserving numerical accuracy. Theoretical analysis establishes rigorous error bounds and guarantees on parallel speedup. Extensive experiments across nine representative dynamical systems—including stiff, chaotic, and high-dimensional cases—demonstrate that nnGParareal significantly outperforms both GParareal and classical Parareal: it achieves faster convergence, higher parallel efficiency, and superior strong scaling behavior.

In large-scale Gaussian process (GP) hyperparameter optimization, iterative linear solvers—such as conjugate gradient (CG)—induce inefficiency in computing gradients of the marginal likelihood due to repeated, costly matrix-vector operations. Method: We propose a general-purpose optimization framework integrating pathwise gradient estimation, solver warm-starting, and budget-aware early stopping. The framework is agnostic to the underlying iterative solver and supports CG, alternating projections, and stochastic gradient descent. Contribution/Results: Our approach substantially alleviates the accuracy–efficiency trade-off in gradient estimation. Experiments demonstrate up to 72× speedup over standard CG when solving to full convergence. Under early stopping, the average residual norm drops to one-seventh of that achieved by baseline methods, significantly shortening hyperparameter optimization time while preserving convergence stability and gradient estimation accuracy.

Existing probabilistic ODE solvers struggle to simultaneously achieve numerical stability and scalability when applied to stiff, high-dimensional problems. This work proposes two complementary strategies to address this challenge: first, a matrix-free update mechanism leveraging Jacobian-vector products, iterative linear solvers, and stochastic covariance estimation to attain linear computational complexity while ensuring numerical stability; second, an iterative re-linearization scheme that reformulates the solver into a fully implicit form, further enhancing stability without compromising scalability. The resulting method constitutes the first probabilistic ODE solver that combines high stability with linear scalability, demonstrating substantial improvements over state-of-the-art approaches across multiple benchmark stiff, high-dimensional ODE systems.

Conventional ODE parameter estimation neglects numerical uncertainty in ODE solvers, leading to unreliable Bayesian inference. Method: We propose the first end-to-end probabilistic parameter inference framework for ODEs, featuring a lightweight, linear-complexity probabilistic ODE solver that explicitly incorporates numerical uncertainty into the Bayesian inference pipeline—integrated with automatic differentiation, just-in-time compilation, variational inference, and MCMC sampling. Contributions/Results: (1) We release the first open-source Python library enabling scalable probabilistic ODE solving; (2) Empirical evaluation across diverse ODE systems demonstrates faster convergence, higher estimation accuracy, and improved posterior calibration compared to deterministic baselines; (3) Inference speed improves by an order of magnitude, significantly enhancing the reliability and practicality of large-scale ODE parameter inference.

High-accuracy deterministic solvers incur prohibitive computational costs when linear systems must be solved repeatedly or only partial solution information is required. This work proposes Monte Carlo Boosting (MC Boosting), an efficient framework for solving linear systems by combining random-walk-based estimators with sequential residual correction. The method unifies Neumann series representations, forward/adjoint estimators, and Halton-sequence-based correction within a single coherent framework, and extends naturally to overdetermined systems, least-squares problems, and iterative algorithms such as IRLS and EM/ECM. By integrating subsampling, sequential Monte Carlo techniques, and variants of Jacobi and Gauss–Seidel iterations, MC Boosting demonstrates significant performance advantages over conventional solvers across diverse extended settings, establishing itself as an efficient computational primitive for modern statistical learning pipelines.

This work addresses the NP-hard problem of penalized least trimmed squares (LTS) regression in robust statistics, for which existing mixed-integer optimization approaches struggle to scale to large datasets. The authors propose a novel formulation that explicitly embeds the arrangement structure of hyperplanes into a perspective reformulation and develop a tailored branch-and-bound algorithm that leverages first-order methods to efficiently solve node relaxations. When the feature dimension is fixed, the method guarantees that the size of the branch-and-bound tree grows polynomially with the sample size, substantially improving both theoretical and practical scalability. Experiments on synthetic data with 5,000 samples and 20 features demonstrate that the proposed approach achieves a 1% optimality gap within one minute, whereas current methods fail to converge within an hour, thereby significantly expanding the tractable scale of exact robust regression.

This work addresses the challenge of parameter optimization in computer model calibration, where the goal is to minimize discrepancies between multidimensional model outputs and observed data. The authors propose a novel root-finding paradigm that reformulates calibration as a root-search problem by constructing signed-residual-based Kriging or stochastic Kriging surrogates. Their approach integrates a sequential search space reduction strategy with a new acquisition function compatible with first-order optimizers. Notably, it guarantees algorithmic robustness even when roots may not exist—a scenario unaddressed by prior methods. Empirical evaluations demonstrate that the proposed method significantly outperforms conventional calibration techniques across both data-driven and physics-based modeling tasks, achieving higher computational efficiency while enhancing solution robustness.

This work addresses the critical challenge that modern GPU-accelerated linear programming solvers—such as cuPDLP, which is based on the primal-dual hybrid gradient (PDHG) algorithm—exhibit performance highly sensitive to hyperparameters, yet lack tuning methods with provable generalization guarantees. For the first time, this study establishes structural relationships between hyperparameters and solution trajectories for multiple adaptive techniques in complex first-order LP solvers, including preconditioning, restart strategies, and smoothed weight updates. By integrating convergence analysis of PDHG with a model of structural sensitivity, the authors propose a data-driven hyperparameter learning framework that offers theoretical generalization guarantees under polynomial sample complexity. Experimental results demonstrate that the framework significantly enhances solver efficiency across diverse problem instances.

Second-order optimization methods remain underutilized due to their implementation complexity, sensitivity to hyperparameters, and lack of composable interfaces. This work proposes Somax, a composable framework natively compatible with Optax that explicitly exposes curvature operators, estimators, and linear solvers as first-class components, enabling flexible composition and efficient reuse. By decoupling computation graph construction from execution through static scheduling and leveraging JIT compilation, Somax encapsulates curvature-aware training into a single-step operation that seamlessly integrates standard optimization techniques such as momentum and weight decay. Systematic experiments demonstrate that well-chosen module compositions significantly improve convergence speed and scalability, while static scheduling effectively reduces per-step computational overhead.