Robust Estimation

Tensegrity robots offer compliance and adaptability, but their nonlinear, and underconstrained dynamics make state estimation challenging. Reliable continuous-time estimation of all rigid links is crucial for closed-loop control, system identification, and machine learning; however, conventional methods often fall short. This paper proposes a two-stage approach for robust state or trajectory estimation (i.e., filtering or smoothing) of a cable-driven tensegrity robot. For online state estimation, this work introduces a factor-graph-based method, which fuses measurements from an RGB-D camera with on-board cable length sensors. To the best of the authors' knowledge, this is the first application of factor graphs in this domain. Factor graphs are a natural choice, as they exploit the robot's structural properties and provide effective sensor fusion solutions capable of handling nonlinearities in practice. Both the Mahalanobis distance-based clustering algorithm, used to handle noise, and the Chebyshev polynomial method, used to estimate the most probable velocities and intermediate states, are shown to perform well on simulated and real-world data, compared to an ICP-based algorithm. Results show that the approach provides high fidelity, continuous-time state and trajectory estimates for complex tensegrity robot motions.

Output-length prediction is important for efficient LLM serving, as it directly affects batching, memory reservation, and scheduling. For prompt-only length prediction, most existing methods use a one-shot sampled length as the label, implicitly treating each prompt as if it had one true target length. We show that this is unreliable: even under a fixed model and decoding setup, the same prompt induces a prompt-conditioned output length distribution, not a deterministic scalar, and this distribution is consistent with heavy-tailed behavior. Motivated by this, we cast length prediction as robust estimation from heavy-tailed prompt-conditioned length distributions. We propose prompt-conditioned length distribution (ProD) methods, which construct training targets from multiple independent generations of the same prompt. Two variants are developed to reuse the served LLM's hidden states: \mbox{ProD-M}, which uses a median-based target for robust point prediction, and ProD-D, which uses a distributional target that preserves prompt-conditioned uncertainty. We provide theoretical justifications by analyzing the estimation error under a surrogate model. Experiments across diverse scenarios show consistent gains in prediction quality.

We study mean estimation of a random vector $X$ in a distributed parameter-server-worker setup. Worker $i$ observes samples of $a_i^\top X$, where $a_i^\top$ is the $i$th row of a known sensing matrix $A$. The key challenges are adversarial measurements and asynchrony: a fixed subset of workers may transmit corrupted measurements, and workers are activated asynchronously--only one is active at any time. In our previous work, we proposed a two-timescale $\ell_1$-minimization algorithm and established asymptotic recovery under a null-space-property-like condition on $A$. In this work, we establish tight non-asymptotic convergence rates under the same null-space-property-like condition. We also identify relaxed conditions on $A$ under which exact recovery may fail but recovery of a projected component of $\mathbb{E}[X]$ remains possible. Overall, our results provide a unified finite-time characterization of robustness, identifiability, and statistical efficiency in distributed linear estimation with adversarial workers, with implications for network tomography and related distributed sensing problems.

Ordinary differential equations (ODEs) provide a powerful framework for modeling dynamic systems arising in a wide range of scientific domains. However, most existing ODE methods focus on a single system, and do not adequately address the problem of learning shared patterns from multiple heterogeneous dynamic systems. In this article, we propose a novel distributionally robust learning approach for modeling heterogeneous ODE systems. Specifically, we construct a robust dynamic system by maximizing a worst-case reward over an uncertainty class formed by convex combinations of the derivatives of trajectories. We show the resulting estimator admits an explicit weighted average representation, where the weights are obtained from a quadratic optimization that balances information across multiple data sources. We further develop a bi-level stabilization procedure to address potential instability in estimation. We establish rigorous theoretical guarantees for the proposed method, including consistency of the stabilized weights, error bound for robust trajectory estimation, and asymptotical validity of pointwise confidence interval. We demonstrate that the proposed method considerably improves the generalization performance compared to the alternative solutions through both extensive simulations and the analysis of an intracranial electroencephalogram data.

In this work, we propose an adaptive robust loss function framework for MHE, integrating an adaptive robust loss function to reduce the impact of outliers with a regularization term that avoids naive solutions. The proposed approach prioritizes the fitting of uncontaminated data and downweights the contaminated ones. A tuning parameter is incorporated into the framework to control the shape of the loss function for adjusting the estimator's robustness to outliers. The simulation results demonstrate that adaptation occurs in just a few iterations, whereas the traditional behaviour $\mathrm{L_2}$ predominates when the measurements are free of outliers.

This paper focuses on the state estimation problem in distributed sensor networks, where intermittent packet dropouts, corrupted observations, and unknown noise covariances coexist. To tackle this challenge, we formulate the joint estimation of system states, noise parameters, and network reliability as a Bayesian variational inference problem, and propose a novel variational Bayesian adaptive Kalman filter (VB-AKF) to approximate the joint posterior probability densities of the latent parameters. Unlike existing AKF that separately handle missing data and measurement outliers, the proposed VB-AKF adopts a dual-mask generative model with two independent Bernoulli random variables, explicitly characterizing both observable communication losses and latent data authenticity. Additionally, the VB-AKF integrates multiple concurrent multiple observations into the adaptive filtering framework, which significantly enhances statistical identifiability. Comprehensive numerical experiments verify the effectiveness and asymptotic optimality of the proposed method, showing that both parameter identification and state estimation asymptotically converge to the theoretical optimal lower bound with the increase in the number of sensors.

Estimation of heterogeneous long-term treatment effects (HLTEs) is relevant for personalized decision-making in marketing, economics, and medicine, where short-term observational datasets are often combined with long-term observational datasets. However, HLTE estimation is challenging due to limited overlap in treatment assignments or in long-term outcomes for certain subpopulations, which can lead to unstable HLTE estimates with large finite-sample variance. To address this challenge, we introduce the LT-O-learners (Long-Term Orthogonal Learners), a set of novel orthogonal learners for HLTE estimation in the canonical HLTE setting with surrogacy. The key idea of our LT-O-learners is to retarget the loss via custom overlap weights that downweight low-overlap samples. We show that the retargeted loss recovers the true HLTE pointwise and satisfies Neyman-orthogonality. We further prove two key theoretical results: (i) The nuisance error enters the error bound only through higher-order terms, which means our learners are robust to nuisance estimation error. (ii) Under a linear function class, the retargeting effectively controls the asymptotic variance of the HLTE estimator via the overlap weights in low-overlap regimes. We conduct experiments on synthetic and real-world datasets to confirm the theoretical properties of our LT-O-learners, particularly robustness in low-overlap regimes. To our knowledge, ours are the first orthogonal learners for HLTE estimation robust to low overlap in long-term settings.

The reliable operation of Unmanned Underwater Vehicle (UUV) clusters is highly dependent on continuous acoustic communication. However, this communication method is highly susceptible to intermittent interruptions. When communication outages occur, standard state estimators such as the Unscented Kalman Filter (UKF) will be forced to make open-loop predictions. If the environment contains unmodeled dynamic factors, such as unknown ocean currents, this estimation error will grow rapidly, which may eventually lead to mission failure. To address this critical issue, this paper proposes a Variational History Distillation (VHD) approach. VHD regards trajectory prediction as an approximate Bayesian reasoning process, which links a standard motion model based on physics with a pattern extracted directly from the past trajectory of the UUV. This is achieved by synthesizing ``virtual measurements'' distilled from historical trajectories. Recognizing that the reliability of extrapolated historical trends degrades over extended prediction horizons, an adaptive confidence mechanism is introduced. This mechanism allows the filter to gradually reduce the trust of virtual measurements as the communication outage time is extended. Extensive Monte Carlo simulations in a high-fidelity environment demonstrate that the proposed method achieves a 91% reduction in prediction Root Mean Square Error (RMSE), reducing the error from approximately 170 m to 15 m during a 40-second communication outage. These results demonstrate that VHD can maintain robust state estimation performance even under complete communication loss.

Parameter estimation in robotics and computer vision faces formidable challenges from both outlier contamination and nonconvex optimization landscapes. While M-estimation addresses the problem of outliers through robust loss functions, it creates severely nonconvex problems that are difficult to solve globally. Adaptive reweighting schemes provide one particularly appealing strategy for implementing M-estimation in practice: these methods solve a sequence of simpler weighted least squares (WLS) subproblems, enabling both the use of standard least squares solvers and the recovery of higher-quality estimates than simple local search. However, adaptive reweighting still crucially relies upon solving the inner WLS problems effectively, a task that remains challenging in many robotics applications due to the intrinsic nonconvexity of many common parameter spaces (e.g. rotations and poses). In this paper, we show how one can easily implement adaptively reweighted M-estimators with certifiably correct solvers for the inner WLS subproblems using only fast local optimization over smooth manifolds. Our approach exploits recent work on certifiable factor graph optimization to provide global optimality certificates for the inner WLS subproblems while seamlessly integrating into existing factor graph-based software libraries and workflows. Experimental evaluation on pose-graph optimization and landmark SLAM tasks demonstrates that our adaptively reweighted certifiable estimation approach provides higher-quality estimates than alternative local search-based methods, while scaling tractably to realistic problem sizes.

Estimating time-varying reproduction numbers from epidemic incidence data is a central task in infectious disease surveillance, yet it poses an inherently ill-posed inverse problem. Existing approaches often rely on strong structural assumptions derived from epidemiological models, which can limit their ability to adapt to non-stationary transmission dynamics induced by interventions or behavioral changes, leading to delayed detection of regime shifts and degraded estimation accuracy. In this work, we propose a Conditional Inverse Reproduction Learning framework (CIRL) that addresses the inverse problem by learning a {conditional mapping} from historical incidence patterns and explicit time information to latent reproduction numbers. Rather than imposing strongly enforced parametric constraints, CIRL softly integrates epidemiological structure with flexible likelihood-based statistical modeling, using the renewal equation as a forward operator to enforce dynamical consistency. The resulting framework combines epidemiologically grounded constraints with data-driven temporal representations, producing reproduction number estimates that are robust to observation noise while remaining responsive to abrupt transmission changes and zero-inflated incidence observations. Experiments on synthetic epidemics with controlled regime changes and real-world SARS and COVID-19 data demonstrate the effectiveness of the proposed approach.

Robotic affordance estimation is challenging due to visual, geometric, and semantic ambiguities in sensory input. We propose a method that disambiguates these signals using two coupled recursive estimators for sub-aspects of affordances: graspable and movable regions. Each estimator encodes property-specific regularities to reduce uncertainty, while their coupling enables bidirectional information exchange that focuses attention on regions where both agree, i.e., affordances. Evaluated on a real-world dataset, our method outperforms three recent affordance estimators (Where2Act, Hands-as-Probes, and HRP) by 308%, 245%, and 257% in precision, and remains robust under challenging conditions such as low light or cluttered environments. Furthermore, our method achieves a 70% success rate in our real-world evaluation. These results demonstrate that coupling complementary estimators yields precise, robust, and embodiment-appropriate affordance predictions.

Estimating multi-component T2 relaxation distributions from Multi-Echo Spin Echo (MESE) MRI is a severely ill-posed inverse problem, traditionally solved using regularized non-negative least squares (NNLS). In abdominal imaging, particularly the pancreas, low SNR and residual uncorrelated noise challenge classical solvers and deterministic deep learning models. We introduce a bootstrap-based inference framework for robust distributional T2 estimation that performs stochastic resampling of the echo train and aggregates predictions across multiple subsets. This treats the acquisition as a distribution rather than a fixed input, yielding variance-reduced, physically consistent estimates and converting deterministic relaxometry networks into probabilistic ensemble predictors. Applied to the P2T2 architecture, our method uses inference-time bootstrapping to smooth noise artifacts and enhance fidelity to the underlying relaxation distribution. Noninvasive pancreatic evaluation is limited by location and biopsy risks, highlighting the need for biomarkers capable of capturing early pathophysiological changes. In type 1 diabetes (T1DM), progressive beta-cell destruction begins years before overt hyperglycemia, yet current imaging cannot assess early islet decline. We evaluate clinical utility via a test-retest reproducibility study (N=7) and a T1DM versus healthy differentiation task (N=8). Our approach achieves the lowest Wasserstein distances across repeated scans and superior sensitivity to physiology-driven shifts in the relaxation-time distribution, outperforming NNLS and deterministic deep learning baselines. These results establish inference-time bootstrapping as an effective enhancement for quantitative T2 relaxometry in low-SNR abdominal imaging.

Standard sequential inference architectures are compromised by a normalizability crisis when confronted with extreme, structured outliers. By operating on unbounded parameter spaces, state-of-the-art estimators lack the intrinsic geometry required to appropriately sever anomalies, resulting in unbounded covariance inflation and mean divergence. This paper resolves this structural failure by analyzing the abstraction sequence of inference at the meta-prior level (S_2). We demonstrate that extremizing the action over an infinite-dimensional space requires a non-parametric field anchored by a pre-prior, as a uniform volume element mathematically does not exist. By utilizing strictly invariant Delta (or ν) Information Separations on the statistical manifold, we physically truncate the infinite tails of the spatial distribution. When evaluated as a Radon-Nikodym derivative against the base measure, the active parameter space compresses into a strictly finite, normalizable probability droplet. Empirical benchmarks across three domains--LiDAR maneuvering target tracking, high-frequency cryptocurrency order flow, and quantum state tomography--demonstrate that this bounded information geometry analytically truncates outliers, ensuring robust estimation without relying on infinite-tailed distributional assumptions.

Random sample consensus (RANSAC), which is based on a repetitive sampling from a given dataset, is one of the most popular robust estimation methods. In this study, an energy-based model (EBM) for robust estimation that has a similar scheme to RANSAC, energy-based RANSAC (EB-RANSAC), is proposed. EB-RANSAC is applicable to a wide range of estimation problems similar to RANSAC. However, unlike RANSAC, EB-RANSAC does not require a troublesome sampling procedure and has only one hyperparameter. The effectiveness of EB-RANSAC is numerically demonstrated in two applications: a linear regression and maximum likelihood estimation.

Principal Component Analysis (PCA) is a cornerstone of dimensionality reduction, yet its classical formulation relies critically on second-order moments and is therefore fragile in the presence of heavy-tailed data and impulsive noise. While numerous robust PCA variants have been proposed, most either assume finite variance, rely on sparsity-driven decompositions, or address robustness through surrogate loss functions without a unified treatment of infinite-variance models. In this paper, we study PCA for high-dimensional data generated according to a superstatistical dependent model of the form $\mathbf{X} = A^{1/2}\mathbf{G}$, where $A$ is a positive random scalar and $\mathbf{G}$ is a Gaussian vector. This framework captures a wide class of heavy-tailed distributions, including multivariate $t$ and sub-Gaussian $α$-stable laws. We formulate PCA under a logarithmic loss, which remains well defined even when moments do not exist. Our main theoretical result shows that, under this loss, the principal components of the heavy-tailed observations coincide with those obtained by applying standard PCA to the covariance matrix of the underlying Gaussian generator. Building on this insight, we propose robust estimators for this covariance matrix directly from heavy-tailed data and compare them with the empirical covariance and Tyler's scatter estimator. Extensive experiments, including background denoising tasks, demonstrate that the proposed approach reliably recovers principal directions and significantly outperforms classical PCA in the presence of heavy-tailed and impulsive noise, while remaining competitive under Gaussian noise.

The autocovariance least squares (ALS) method is a computationally efficient approach for estimating noise covariances in Kalman filters without requiring specific noise models. However, conventional ALS and its variants rely on the classic least mean squares (LMS) criterion, making them highly sensitive to measurement outliers and prone to severe performance degradation. To overcome this limitation, this paper proposes a novel outlier-robust ALS algorithm, termed ALS-IRLS, based on the iteratively reweighted least squares (IRLS) framework. Specifically, the proposed approach introduces a two-tier robustification strategy. First, an innovation-level adaptive thresholding mechanism is employed to filter out heavily contaminated data. Second, the outlier-contaminated autocovariance is formulated using an $ε$-contamination model, where the standard LMS criterion is replaced by the Huber cost function. The IRLS method is then utilized to iteratively adjust data weights based on estimation deviations, effectively mitigating the influence of residual outliers. Comparative simulations demonstrate that ALS-IRLS reduces the root-mean-square error (RMSE) of noise covariance estimates by over two orders of magnitude compared to standard ALS. Furthermore, it significantly enhances downstream state estimation accuracy, outperforming existing outlier-robust Kalman filters and achieving performance nearly equivalent to the ideal Oracle lower bound in the presence of noisy and anomalous data.

The Unscented Kalman Filter (UKF) is a ubiquitous tool for nonlinear state estimation; however, its performance is limited by the static parameterization of the Unscented Transform (UT). Conventional weighting schemes, governed by fixed scaling parameters, assume implicit Gaussianity and fail to adapt to time-varying dynamics or heavy-tailed measurement noise. This work introduces the Meta-Adaptive UKF (MA-UKF), a framework that reformulates sigma-point weight synthesis as a hyperparameter optimization problem addressed via memory-augmented meta-learning. Unlike standard adaptive filters that rely on instantaneous heuristic corrections, our approach employs a Recurrent Context Encoder to compress the history of measurement innovations into a compact latent embedding. This embedding informs a policy network that dynamically synthesizes the mean and covariance weights of the sigma points at each time step, effectively governing the filter's trust in the prediction versus the measurement. By optimizing the system end-to-end through the filter's recursive logic, the MA-UKF learns to maximize tracking accuracy while maintaining estimation consistency. Numerical benchmarks on maneuvering targets demonstrate that the MA-UKF significantly outperforms standard baselines, exhibiting superior robustness to non-Gaussian glint noise and effective generalization to out-of-distribution (OOD) dynamic regimes unseen during training.

Robust estimation of object poses in robotic manipulation is often addressed using foundational general estimators, that aim to handle diverse error sources naively within a single model. Still, they struggle due to environmental uncertainties, while requiring long inference times and heavy computation. In contrast, we propose a modular, uncertainty-aware framework that attributes pose estimation errors to specific error sources and applies targeted mitigation strategies only when necessary. Instantiated with Iterative Closest Point (ICP) as a simple and lightweight pose estimator, we leverage our framework for real-world robotic grasping tasks. By decomposing pose estimation into failure detection, error attribution, and targeted recovery, we significantly improve the robustness of ICP and achieve competitive performance compared to foundation models, while relying on a substantially simpler and faster pose estimator.

Cooperative localization is essential for swarm applications like collaborative exploration and search-and-rescue missions. However, maintaining real-time capability, robustness, and computational efficiency on resource-constrained platforms presents significant challenges. To address these challenges, we propose D-GVIO, a buffer-driven and fully decentralized GNSS-Visual-Inertial Odometry (GVIO) framework that leverages a novel buffering strategy to support efficient and robust distributed state estimation. The proposed framework is characterized by four core mechanisms. Firstly, through covariance segmentation, covariance intersection and buffering strategy, we modularize propagation and update steps in distributed state estimation, significantly reducing computational and communication burdens. Secondly, the left-invariant extended Kalman filter (L-IEKF) is adopted for information fusion, which exhibits superior state estimation performance over the traditional extended Kalman filter (EKF) since its state transition matrix is independent of the system state. Thirdly, a buffer-based re-propagation strategy is employed to handle delayed measurements efficiently and accurately by leveraging the L-IEKF, eliminating the need for costly re-computation. Finally, an adaptive buffer-driven outlier detection method is proposed to dynamically cull GNSS outliers, enhancing robustness in GNSS-challenged environments.

We study the basic task of mean estimation in the presence of mean-shift contamination. In the mean-shift contamination model, an adversary is allowed to replace a small constant fraction of the clean samples by samples drawn from arbitrarily shifted versions of the base distribution. Prior work characterized the sample complexity of this task for the special cases of the Gaussian and Laplace distributions. Specifically, it was shown that consistent estimation is possible in these cases, a property that is provably impossible in Huber's contamination model. An open question posed in earlier work was to determine the sample complexity of mean estimation in the mean-shift contamination model for general base distributions. In this work, we study and essentially resolve this open question. Specifically, we show that, under mild spectral conditions on the characteristic function of the (potentially multivariate) base distribution, there exists a sample-efficient algorithm that estimates the target mean to any desired accuracy. We complement our upper bound with a qualitatively matching sample complexity lower bound. Our techniques make critical use of Fourier analysis, and in particular introduce the notion of a Fourier witness as an essential ingredient of our upper and lower bounds.

We study high-dimensional mean estimation in a collaborative setting where data is contributed by $N$ users in batches of size $n$. In this environment, a learner seeks to recover the mean $μ$ of a true distribution $P$ from a collection of sources that are both statistically heterogeneous and potentially malicious. We formalize this challenge through a double corruption landscape: an $\varepsilon$-fraction of users are entirely adversarial, while the remaining ``good'' users provide data from distributions that are related to $P$, but deviate by a proximity parameter $α$. Unlike existing work on the untrusted batch model, which typically measures this deviation via total variation distance in discrete settings, we address the continuous, high-dimensional regime under two natural variants for deviation: (1) good batches are drawn from distributions with a mean-shift of $\sqrtα$, or (2) an $α$-fraction of samples within each good batch are adversarially corrupted. In particular, the second model presents significant new challenges: in high dimensions, unlike discrete settings, even a small fraction of sample-level corruption can shift empirical means and covariances arbitrarily. We provide two Sum-of-Squares (SoS) based algorithms to navigate this tiered corruption. Our algorithms achieve the minimax-optimal error rate $O(\sqrt{\varepsilon/n} + \sqrt{d/nN} + \sqrtα)$, demonstrating that while heterogeneity $α$ represents an inherent statistical difficulty, the influence of adversarial users is suppressed by a factor of $1/\sqrt{n}$ due to the internal averaging afforded by the batch structure.

Conditional Value-at-Risk (CVaR) is a widely used risk-sensitive objective for learning under rare but high-impact losses, yet its statistical behavior under heavy-tailed data remains poorly understood. Unlike expectation-based risk, CVaR depends on an endogenous, data-dependent quantile, which couples tail averaging with threshold estimation and fundamentally alters both generalization and robustness properties. In this work, we develop a learning-theoretic analysis of CVaR-based empirical risk minimization under heavy-tailed and contaminated data. We establish sharp, high-probability generalization and excess risk bounds under minimal moment assumptions, covering fixed hypotheses, finite and infinite classes, and extending to $β$-mixing dependent data; we further show that these rates are minimax optimal. To capture the intrinsic quantile sensitivity of CVaR, we derive a uniform Bahadur-Kiefer type expansion that isolates a threshold-driven error term absent in mean-risk ERM and essential in heavy-tailed regimes. We complement these results with robustness guarantees by proposing a truncated median-of-means CVaR estimator that achieves optimal rates under adversarial contamination. Finally, we show that CVaR decisions themselves can be intrinsically unstable under heavy tails, establishing a fundamental limitation on decision robustness even when the population optimum is well separated. Together, our results provide a principled characterization of when CVaR learning generalizes and is robust, and when instability is unavoidable due to tail scarcity.

Phased-array Bluetooth systems have emerged as a low-cost alternative for performing aided inertial navigation in GNSS-denied use cases such as warehouse logistics, drone landings, and autonomous docking. Basing a navigation system off of commercial-off-the-shelf components may reduce the barrier of entry for phased-array radio navigation systems, albeit at the cost of significantly noisier measurements and relatively short feasible range. In this paper, we compare robust estimation strategies for a factor graph optimisation-based estimator using experimental data collected from multirotor drone flight. We evaluate performance in loss-of-GNSS scenarios when aided by Bluetooth angular measurements, as well as range or barometric pressure.

The problem of linear predictions has been extensively studied for the past century under pretty generalized frameworks. Recent advances in the robust statistics literature allow us to analyze robust versions of classical linear models through the prism of Median of Means (MoM). Combining these approaches in a piecemeal way might lead to ad-hoc procedures, and the restricted theoretical conclusions that underpin each individual contribution may no longer be valid. To meet these challenges coherently, in this study, we offer a unified robust framework that includes a broad variety of linear prediction problems on a Hilbert space, coupled with a generic class of loss functions. Notably, we do not require any assumptions on the distribution of the outlying data points ($\mathcal{O}$) nor the compactness of the support of the inlying ones ($\mathcal{I}$). Under mild conditions on the dual norm, we show that for misspecification level $ε$, these estimators achieve an error rate of $O(\max\left\{|\mathcal{O}|^{1/2}n^{-1/2}, |\mathcal{I}|^{1/2}n^{-1} \right\}+ε)$, matching the best-known rates in literature. This rate is slightly slower than the classical rates of $O(n^{-1/2})$, indicating that we need to pay a price in terms of error rates to obtain robust estimates. Additionally, we show that this rate can be improved to achieve so-called "fast rates" under additional assumptions.