nonlinear least squares

This work addresses the joint estimation of states (e.g., poses, 3D points) and their noise covariance matrices under Gaussian noise. We provide the first rigorous proof that the joint maximum a posteriori (MAP) / maximum likelihood estimation (MLE) objective is convex, and derive a closed-form analytical solution for the covariance matrix. Leveraging this theoretical insight, we propose two provably convergent joint estimation algorithms: (i) a direct convex optimization method, and (ii) an efficient algorithm integrating iteratively reweighted least squares. Compared to methods assuming fixed covariances or employing expectation-maximization (EM)-type frameworks, our approach achieves significantly improved pose and map accuracy—and enhanced robustness—across diverse robotics and visual localization tasks, including SLAM.

This paper addresses nonlinear state estimation under unknown dynamic and observation models. We propose Neural Kalman Maximum Likelihood Estimation (NK-MLE), a supervised learning framework that jointly learns neural-network-parameterized nonlinear dynamics and observation functions alongside Gaussian noise covariance matrices. Our key innovation is a coordinate ascent optimization strategy that alternately updates neural network parameters and noise parameters, enabling end-to-end differentiability within a Kalman filtering pipeline. Upon training, the learned model can be seamlessly integrated into standard extended or unscented Kalman filters as a drop-in replacement. Experiments across multiple nonlinear dynamical systems demonstrate that NK-MLE significantly improves estimation accuracy and robustness—particularly under model mismatch and uncertain noise statistics—outperforming conventional approaches in both nominal and challenging regimes.

This work addresses large-scale spatiotemporal systems with unknown or missing sensor models by proposing an inverse sensing architecture that synthesizes measurement likelihoods under prescribed accuracy constraints. The method minimizes information injection into the dynamic prior while ensuring the synthesized likelihood satisfies a specified error bound. Its core innovation lies in a unified maximum-entropy posterior framework for likelihood synthesis, which leverages relative entropy minimization and Radon–Nikodym derivatives to accommodate diverse discrepancy measures—including Wasserstein distance, maximum mean discrepancy (MMD), and f-divergences—and establishes a direct mapping between accuracy budgets and physical sensor configurations. Combining particle filtering with convex optimization, experiments validate the effectiveness of accuracy-constrained synthesis across four discrepancy measures, reveal how the choice of measure influences both the quantity and spatial distribution of injected information, and demonstrate successful distillation of nonparametric likelihoods into parametric forms.

In robot state estimation, anisotropic (matrix-weighted) noise causes conventional convex relaxations to lose tightness, thereby invalidating certification guarantees. Method: This paper identifies the fundamental mechanism by which matrix weighting undermines the tightness of semidefinite relaxation (SDR), establishes a theoretical link between posterior uncertainty and the certificate matrix, and proposes a novel paradigm—incorporating geometric redundancy constraints—to restore tightness. It further constructs the first tightable semidefinite programming (SDP) formulation for matrix-weighted SLAM. Contribution/Results: We theoretically prove that the proposed SDP admits a tightness bound under low-noise conditions. Extensive simulations and real-world experiments demonstrate that redundancy constraints significantly improve tightness rates; moreover, the new formulation maintains tightness even under extremely low noise, outperforming scalar-weighted approaches in both certification reliability and estimation accuracy.

This paper addresses constrained stochastic nonlinear optimization problems arising in online statistical inference. We propose Sketch-StoSQP, a sketched stochastic sequential quadratic programming method. Our key contributions are threefold: (i) We establish, for the first time, the asymptotic normality of StoSQP iterates under controllable, non-vanishing approximation errors—ensuring stable per-iteration computational complexity; (ii) We design a plug-and-play covariance estimator enabling immediate statistical inference without algorithmic modification; (iii) We prove that the scaled residual sequence converges in distribution to a non-degenerate zero-mean Gaussian. Empirical evaluation on the CUTEst benchmark and constrained regression tasks demonstrates both statistical validity—accurate coverage rates and well-calibrated confidence intervals—and computational efficiency—constant per-iteration cost and significant overall speedup.

Existing Bayesian optimization methods lack theoretical guarantees for adaptive data acquisition in nonlinearly parameterized models. This work proposes an analytical framework based on the reproducing kernel Hilbert space (RKHS) induced by kernels over the parameter space, integrated with regularized convex loss minimization, to establish a unified confidence bound theory for widely used nonlinear surrogate models. For the first time, this framework provides rigorous convergence guarantees for nonlinearly parameterized models under adaptive sampling, enabling a variety of novel acquisition strategies—including stochastic regularization and randomized model maximization—and substantially broadening the theoretical applicability of Bayesian optimization.

This work addresses key challenges in nonlinear statistical inverse problems—namely, intractable likelihood evaluation, nonconvex optimization landscapes, and slow mixing of MCMC samplers—by introducing plug-in and PDE-penalized M-estimators. These estimators relax the underlying PDE constraints to transform the original problem into a conditionally convex or nested quadratic optimization framework, thereby circumventing the need for explicit forward map evaluations or iterative PDE solvers. The proposed approach is the first to simultaneously achieve optimal statistical convergence rates and global polynomial-time computability. Moreover, it establishes a generalized stability theory that extends beyond the range of the forward operator. In canonical settings such as Darcy flow inversion, the method attains state-of-the-art statistical rates while yielding an explicit sub-quadratic runtime bound of $o(N^2)$, and further provides high-quality initializations for Bayesian inference.

This work proposes a novel framework that integrates classical particle filtering with learning-based methods to address the high training cost and poor interpretability of end-to-end learning in robotic state estimation. Leveraging the Markov assumption, the approach trains a dynamics model using single-step state transitions and implicitly learns the observation model via denoising score matching, thereby approximating the Bayesian filtering equations step-by-step during inference without requiring end-to-end optimization. A key innovation lies in preserving the modular structure of the filter, which enables flexible incorporation of prior knowledge and external sensor models without retraining. Experiments demonstrate that the method achieves accuracy comparable to well-tuned end-to-end baselines in simulation while significantly reducing training complexity and exhibiting superior generalization and compositional capabilities.

This work addresses the lack of robustness in ordinary least squares when sparse, large-magnitude outliers are present. The authors propose a maximum-entropy-based weighted least squares framework, wherein weights are interpreted as a discrete probability distribution and determined by maximizing Shannon entropy subject to a mean squared error constraint. This approach automatically downweights outliers while preserving fidelity to inliers. Innovatively treating the mean squared error as a tunable control parameter, the method leverages the principle of maximum entropy to select the least biased weight distribution. The paper establishes the theoretical existence of a locally unique, globally continuable smooth solution branch emanating from the ordinary least squares solution. Further analysis reveals that, in the zero-error limit, the method automatically identifies the largest subset of data consistent with the underlying linear model. Numerical experiments confirm its superior robustness.

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.