arXiv subjects

Qirui Hu

Publications and source records attributed to Qirui Hu.

At least 19 recordsLinked to original sources

arXiv2026

Martingalized U-statistics-based sequential inference: a blessing of degeneracy

Degenerate degree-two U-statistics have non-Gaussian limits, making sequential inference substantially more difficult than in the nondegenerate case. However, row normalization preserves the target parameter and makes the centered numerator a martingale on a harmonic variance clock, at the cost of logarithmic variance inflation. We establish a strong Gaussian approximation for this process and consistently estimate its variance through online updates, allowing spectrum-free calibration. Combining this strong approximation with Gaussian-mixture (GM) and stitched law-of-the-iterated-logarithm (LIL) boundaries yields confidence sequences with asymptotic time-uniform coverage and corresponding sequential tests for null hypotheses under which the kernel is degenerate. We further extend the framework to multi-dimensional vector-valued kernels, yielding Mahalanobis confidence ellipsoids based on consistent covariance estimation. Numerical experiments illustrate sequential type-I error control and favorable detection power in two-sample and independence testing against spectral baselines, and in three-sample homogeneity testing using vector confidence ellipsoids.

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arXiv2026

Sampling Design and Confidence Tubes for Integral Curves on Riemannian Manifolds

We estimate the integral curve of an unknown vector field from noisy tangent vector field on a known compact Riemannian manifold. We consider two sampling settings: independent observations at random locations that have already been collected, and a fixed response budget that allows subsequent locations to be chosen using pilot information. Under suitable regularity assumptions, the passive kernel estimator has curve error of order $n^{-2/(d+3)}$ in probability, with sample size $n$ and manifold dimension $d$. Its functional central limit theorem has a limit characterized by a linear stochastic differential equation. For active sampling, an independent pilot directs one new batch of measurements to neighborhoods of the target curve. Under localization, smoothness and design conditions, this concentration yields expected maximum curve error of order $N^{-1/2}$, including the pilot in the total response budget $N$. Both methods provide simultaneous geodesic confidence tubes with asymptotically exact coverage under mild conditions. Simulations show narrower active tubes in most reported settings, with improved small-sample coverage in some three-dimensional examples. Ocean-drifter observations illustrate mean-flow reconstruction on the sphere, and a canopy-flow LDA application uses retrospective active sampling to quantify lateral and vertical departures of local mean streamlines.

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arXiv2026

Bootstrap Inference for Riemannian Optimization Problems

We develop an online multiplier bootstrap for manifold-valued parameters estimated by averaging the iterates of Riemannian stochastic approximation. The method runs weighted iterations alongside the original algorithm, with each path using the same observations while updating its own state and intrinsic average. The weighted averages provide uncertainty assessments without a plug-in asymptotic covariance estimator. Under local stability and regularity conditions, we establish bootstrap consistency and show that a fixed number of paths suffices for asymptotically valid Student intervals and intrinsic confidence regions. The theory accommodates samplewise nonsmooth scores and identifies a critical oracle-bias regime in which the estimator's limiting distribution shifts, but the centered bootstrap distribution does not. Applications include spherical means, positive-definite matrix means and medians, and principal-subspace estimation. Simulations show coverage close to nominal in regular settings, and an analysis of North Atlantic storm locations illustrates descriptive online inference for a spherical center.

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arXiv2026

Riemannian Newton Methods with Statistical Inference

We develop online statistical inference for Polyak-Ruppert averages of inexact Newton iterates on Riemannian manifolds. A local asymptotic representation separates observation noise, solver randomization, and accumulated implementation bias, and yields one-pass covariance estimators and asymptotically valid Wald regions under local regularity and centering conditions. Independent fresh solves add a positive semidefinite term to the score-sandwich covariance; averaging solver replicas reduces this term in inverse proportion to their number. Slowly varying predictable gains instead cancel from the first order law. Product torus simulations show how Newton scaling shortens an ill-conditioned transient and how retaining solver variation restores coverage. Intrinsic means and principal subspaces provide geometric applications, with a streaming data illustration.

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arXiv2026

Self-Normalized Inference from Riemannian Stochastic Gradient Paths

Riemannian stochastic gradient paths estimate constrained parameters, but confidence regions usually require estimating the Hessian and gradient-noise covariance. We develop self-normalized inference from one path by expressing retained iterates in a common random tangent space. Their average and centered partial sums share an unknown covariance factor that cancels in a Brownian bridge pivot. Under suitable regularity conditions, a joint expansion of the averaged estimator and integrated path controls nonlinear coordinate errors using conditional second moments. The procedure uses fixed-dimensional memory and extends to nondegenerate smooth functionals and alternative local coordinates. Numerical studies show competitive coverage relative to plug-in Wald inference while avoiding Hessian and residual-density estimation.

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arXiv2026

Sharp Critical Rates for Locally Private Distribution Estimation

We determine sharp minimax rates for distribution estimation under pure local differential privacy at critical query smoothness. For the integral probability metric generated by the unit ball of $B^d_{2,q}$, $1\le q\le\infty$, over fixed periodic Sobolev density balls with positive smoothness and interior slack at the uniform density, the expected risk is of order $N^{-1/2}(\log N)^{3/2-1/q}$, where $N=n\varepsilon^2$; expected squared risk has the squared order. The fine index governs aggregation across resolutions that share one private information budget. A single noninteractive release with a measurable proper density output attains these rates, and matching lower bounds allow finite public blackboard interaction and repeated participation under each user's total privacy budget. A known $L^2$ radius permits the same rates without using density smoothness, with output in the $L^2$ density ball. Separately, for positive smooth densities on $[0,1]$, integrated squared CDF risk has minimax order $(\log N)^2/N$, with sharp quantile, Wasserstein, and transport map consequences. The rates hold uniformly for $0<\varepsilon\le1$ as $N\to\infty$, with class parameters fixed.

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arXiv2026

Minimax estimation of linear functionals in the Darcy inverse problem

We study estimation of linear functionals $\int fψ$ of the conductivity $f$ in the Darcy problem $\nabla\cdot(f\nabla u)=g$ from observations of $u$ in Gaussian white noise of level $N^{-1/2}$. When the potential has a single nondegenerate minimum, we determine the minimax rates over Sobolev balls in the linearised experiment. Unless a compatibility condition along the gradient flow holds, the rate is governed by an effective dimension $T$ of the Hessian at the minimum, not by the ambient dimension: it is $N^{-(2s+T)/(4s+8)}$ for $T<4$, has an elbow at $T=4$ with an extra logarithmic factor, and is parametric for $T>4$. In the nonlinear experiment we show that these lower bounds persist when compatibility fails at the reference and $T\le4$, and we prove the minimax rate on classes with isotropic critical Hessians, and the parametric rate near references with $T>4$. In dimension at most four, no nonzero linear functional is uniformly $\sqrt N$-estimable on a Sobolev neighbourhood of the constant conductivity, even when it is regular there.

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arXiv2026

Infinite Quantiles Inference under Local Differential Privacy

We study simultaneous quantile inference under local differential privacy for a number of target probabilities that grows with the sample size. Each user participates once, so learning unknown quantile locations and calibrating their joint uncertainty must share the same private information. We fuse complementary cumulative distribution function (CDF) scores from a single tree report using their exact covariance; the variance at the correct anchor is no larger than that of either score, and anchor error incurs only quadratic excess variance. Three consecutive cohorts then localize the quantiles, estimate their densities, and supply fresh scores for a one-step correction and joint multiplier calibration. Under mild conditions, a uniform representation controls localization and density errors and yields asymptotically valid simultaneous coverage. A fixed-privacy regime permits every polynomially growing target grid. Simulations show consistent same-report precision gains, while comparisons with CDF-band inversion expose the sample cost of learning target locations. Census microdata illustrates how the resulting joint intervals support wage-percentile ratios.

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arXiv2026

FluxVLA Engine: A One-Stop VLA Engineering Platform for Embodied Intelligence

Vision-language-action (VLA) models, world-action models (WAMs), and offline reinforcement learning methods are rapidly expanding the design space of embodied policies, yet turning these algorithms into reliable robot systems remains constrained by fragmented data formats, training stacks, evaluation protocols, inference runtimes, and embodiment-specific interfaces. We present $\mathrm{FluxVLA}$ Engine, an open, configuration-driven platform that turns heterogeneous embodied-policy components into a reproducible data-to-deployment workflow. Rather than introducing another policy model, $\mathrm{FluxVLA}$ standardizes interfaces for datasets, visual-language and world models, action heads, reward- or advantage-weighted learning, distributed training, simulation evaluation, optimized inference, and robot operators. The engine further integrates compositional dual-arm simulation, scalable automatic data generation, and model-decoupled human-in-the-loop rollout, takeover, correction collection, and reward annotation. For responsive physical execution, it combines Real-Time Chunking (RTC) with accelerated inference backends, lightweight remote GPU serving, and configurable trajectory post-processing. Together, these capabilities connect offline learning, simulation validation, online correction, and real-robot execution through shared and auditable contracts. $\mathrm{FluxVLA}$ therefore targets the engineering bottlenecks separating promising embodied-learning algorithms from reproducible evaluation and dependable deployment. Code is available at https://github.com/FluxVLA/FluxVLA

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arXiv2026

Mean estimation in spatial functional data

Functional mean estimation exhibits a sparse-to-dense phase transition as the number of measurements per curve increases. For spatially indexed functional data, however, dependence between nearby curves and increasing spatial sampling density fundamentally alter this transition. We investigate this phenomenon under a unified framework allowing both increasing-domain and infill asymptotics, with irregular temporal observations. Our analysis identifies two distinct sources of estimation error: an observation-level component governed by the total number of measurements and a curve-level component governed by a spatial dependence-adjusted effective sample size. The resulting uniform convergence rates reveal how spatial dependence shifts the temporal phase-transition boundary and how spatial infill can improve estimation through additional measurements without a corresponding increase in effective curve-level information. We establish these results for three spline-based estimators: a pooled estimator of a stationary mean curve, a local-kernel estimator, and a tensor-product estimator of a spatially varying mean surface. The effects of spatial localization on the phase transition are characterized explicitly. Simulations illustrate the interplay between spatial dependence and temporal sampling density, and environmental data applications demonstrate the proposed estimation methods.

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arXiv2026

HWAM: Humanoid World Action Model With Joint State--Action Generation

Humanoid robots are a promising platform for general-purpose manipulation. Recent Vision-Language-Action (VLA) policies learn actions directly from multimodal observations, while World Action Models (WAMs) further incorporate future visual prediction to improve action generation. However, in hierarchical humanoid systems, VLA and WAM policies output reference actions that are subsequently realized through whole-body control, robot dynamics, balance, and contact. This hierarchy creates an action--execution gap: the reference produced by the policy can differ from the motion realized by the robot. Without explicitly modeling the realized body state, future visual prediction must jointly explain scene evolution and discrepancies between reference actions and executed motion, making it difficult to associate an action with its physical outcome. We propose HWAM, a Humanoid World Action Model with joint state--action generation, which makes the robot's post-execution proprioceptive state an explicit prediction target. By jointly generating reference actions and their realized body states, HWAM directly incorporates supervision of executed motion into action learning. HWAM is trained through three complementary conditional paths. The Policy path jointly denoises state--action trajectories conditioned only on current observations, matching deployment conditions. Forward Dynamics Modeling (FDM) predicts future visual observations conditioned on actions and post-execution states, while Inverse Dynamics Modeling (IDM) reconstructs the joint trajectory from visual transitions. Together, these paths connect policy references, realized body motion, and visual outcomes. HWAM achieves the highest success rate among evaluated baselines on three real-robot tasks on the LimX OLI humanoid. On Candy Picking, HWAM achieves a 70.6% success rate, compared with 43.3% for Fast-WAM.

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arXiv2026

Approximation Theorems for High-Dimensional Canonical U-Statistics: Gaussian Chaos and Phase Transition

We study simultaneous inference for maxima of canonical order-two $U$-statistics in high dimension. Degeneracy makes quadratic fluctuations leading, so ordinary Gaussian calibration can fail even after exact variance normalization. We show that the appropriate general target is a joint signed Gaussian quadratic chaos and establish a general approximation result that permits indefinite kernels. The general anti-concentration bound is too crude for high-dimensional inference, and we obtain sharper bounds under additional spectral structure. We also identify a phase transition from a non-Gaussian signed-chaos maximum to its covariance-matched Gaussian counterpart driven by the effective rank. For feasible inference, we propose a Gaussian multiplier bootstrap that avoid estimating eigensystems, and establish its validity. Two applications and extensive numerical simulations further illustrate the scope and practical performance of the proposed framework.

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arXiv2026

Riemannian Simultaneous Inference for Tangent Vector Field Regression

We consider nonparametric tangent vector field regression on a Riemannian manifold without boundary. Because responses at different points lie in different tangent spaces, the proposed kernel estimator first parallel transports nearby responses to the target tangent space and then forms a volume-corrected local average. We first derive its uniform second-order bias, finite-bandwidth covariance, and stochastic rate. For simultaneous inference, the tangent norm is written as a supremum over the unit tangent bundle. Exact covariance whitening gives a unit-variance Gaussian field whose correlation length is of order $h$ along the base manifold and of order one along the fibre. Its local covariance geometry leads to a Gumbel limit with an explicit intrinsic constant. Combining this limit with Gaussian approximation and cross-fitted covariance estimation yields a feasible simultaneous confidence tube for the regression field. We further discuss improved finite-sample inference with bandwidth selection and high-order bias corrections. Simulations on various manifolds support the proposed inference procedure. A randomized reconstruction of global wind data illustrates how the tube's cross-sections describe spatially varying uncertainty.

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arXiv2026

Locally Private Inference for Riemannian Stochastic Optimization

We develop inference for manifold-valued population minimizers when each observation belongs to a different participant and only locally private messages reach the analyst. The method releases randomized tangent gradients and combines them through Riemannian stochastic approximation and Polyak-Ruppert averaging. Directly inserting a private data surrogate into a nonlinear loss can shift its population target, whereas conditional centring of the released gradient preserves the first-order equation. We introduce symmetric-pair regression (SPR) to estimate the asymptotic variance from the same private messages used for point estimation, without holding out participants or requesting a second release. We prove the central limit theorem and consistency of the fully transcript-based sandwich covariance and intrinsic Wald region under local differential privacy. Simulations across various statistical problems and manifolds support the predicted decrease in estimation error and near-nominal coverage under moderate privacy. An application to NHANES anthropometric data illustrates private estimation of a leading body-size direction and its uncertainty.

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arXiv2026

Asymptotic Anytime-Valid Quantile Inference under Local Differential Privacy

Sequential quantile inference is difficult under local differential privacy because every record is randomized before reaching the analyst and the limiting quantile variance depends on an unknown density. We develop an online procedure that combines randomized response with dynamically chained parallel stochastic gradient descent (P-SGD). The resulting Polyak--Ruppert estimator admits a strong Gaussian approximation. A cross-chain quadratic statistic, computed entirely from private iterates, consistently estimates the limiting variance without a separate online density estimator. These results yield asymptotic confidence sequences and, under polynomial chain growth, asymptotic time-uniform coverage. Arm-wise constructions support locally private quantile best-arm identification, time-uniform simple-regret bounds, and sequential A/B tests of quantile treatment effects. Simulations and salary-data analyses illustrate the finite-sample behavior and practical use of the proposed methods.

stat.ME↗

arXiv2026

Multicollinearity-agnostic feature screening for non-Euclidean responses: a factor adjusted approach

In high-dimensional settings, multicollinearity is a pervasive issue that can substantially impair the performance of feature screening methods based on marginal Fréchet regression. Feature screening for non-Euclidean responses becomes unreliable when ultrahigh-dimensional predictors suffer from multicollinearity, because feature-specific signals may be masked by shared latent factors. To mitigate this effect, we propose a Factor adjusted Fréchet sure independence screening procedure. The method first recovers latent common factors from the predictors and then evaluates each feature by the incremental Fréchet coefficient of determination contributed by its idiosyncratic component beyond the common factors. Under regularity conditions, we establish uniform approximation rates for the feasible screening utilities and prove the sure screening and sure ranking properties. Extensive numerical experiments provide compelling empirical support for the validity and effectiveness of our approach, particularly in scenarios with highly correlated covariates. We further illustrate the practical performance of our method through two representative non-Euclidean datasets: the ADNI dataset and the mortality dataset, both with distribution-valued responses.

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arXiv2026

MATCH: Multiplier-Assisted Tests for Conditional Hypotheses in Non-Euclidean Data

We propose a new procedure MATCH (Multiplier-Assisted Tests for Conditional Hypotheses) to test whether the non-Euclidean data match the target model, which is a general framework for significance and specification testing in Fréchet regression. MATCH covers global significance, partial significance, and the adequacy of global Fréchet regression, providing a unified way to compare unrestricted conditional Fréchet means with restricted alternatives. One of the key challenges is that the ordinary held-out loss difference is first-order degenerate under the null: the oracle losses coincide, and plug-in statistics is dominated by nuisance estimation error. MATCH uses sample splitting and independent random multipliers on held-out losses to create a nondegenerate Gaussian leading term without residuals or tangent-space coordinates. To improve data use and stability, we further develop cross-fitted tests and repeated cross-fitting with p-value merging. We establish asymptotic null validity, consistency under fixed alternatives, and local power guarantees. Simulations for distributional, symmetric positive-definite (SPD) matrix-valued, and spherical responses support the theoretical findings, and applications to county-level household income distributions and North Atlantic tropical-cyclone locations demonstrate the practical use of the proposed tests.

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arXiv2026

Locally Private Online Quantile Regression: Estimation and Inference

We study estimation and inference for online quantile regression under a one-report user-level $\eps$-locally differentially private ($\eps$-LDP) protocol. The main difficulty is that the standard quantile-regression estimating-equation contribution couples covariates with a residual comparison, so a server that receives only privatized reports cannot form the usual online update. We address this by developing a finite-alphabet channel in which each user computes the contribution locally, applies support-aware stochastic quantization and randomized response to one selected-block category, and sends one report. A public decoder corrects the randomized-response distortion and reconstructs a server-side estimating-equation input with the correct conditional mean. These decoded inputs are then used in projected Polyak-Ruppert averaging. For fixed finite channel designs, we establish local privacy, decoder unbiasedness, consistency, asymptotic normality, and Hessian-free self-normalized inference for prespecified scalar contrasts. Simulations and a New York City taxi-trip illustration show that the private trajectory approaches the nonprivate online reference as the privacy budget grows and outperforms direct Laplace and face-exponential geometric releases in the reported regimes.

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