Covariance Estimation

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5 papers in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

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Latest papers 30

Oct 5, 2026cs.LG

Joint Precision Neural Networks: Task-Aware Dependency and Predictive Learning

Exploiting meaningful latent structures from data to solve downstream tasks is a fundamental challenge in signal processing and machine learning. While Principal Component Analysis (PCA) and coVariance Neural Networks (VNNs) successfully leverage the covariance matrix to process data, they inherently capture both direct and indirect correlations. The precision matrix (inverse covariance) overcomes this by explicitly encoding conditional independencies, making it largely studied in graphical lasso and graph topology identification. However, finite-sample precision estimates are notoriously unstable, and regularized estimators remain task-agnostic. In this work, our principal contribution is tackling the challenging problem of task-aware graph inference. We propose Precision Neural Networks-Joint (PNN-Joint), a framework that jointly estimates a sparse, statistically grounded precision matrix alongside graph neural network weights via an alternating optimization scheme. As a foundational framework to support this, we introduce Precision Neural Networks (PNNs), a broader class of graph convolutional networks operating on precision estimators, and establish their spectral connections to PCA and VNNs alongside their stability to finite-sample errors. Extensive empirical evaluations on synthetic data, as well as real-world neuroimaging and motion sensor datasets, demonstrate that PNN-Joint yields highly interpretable task-aware graphs, exhibits remarkable robustness in low-data regimes, and consistently achieves the best or second-best performance among competitors on real-world tasks.

Andrea Cavallo, Samuel Rey, Antonio G. Marques +1

INSY Department, Delft University of Technology · Department of Signal Theory and Communications, King Juan Carlos University

Graph Structure Learning Graph Neural Networks Covariance Estimation

Sep 30, 2026stat.ML

Inference for stochastic differential equations driven by weighted sub-fractional Brownian motion using neural networks and the Euler approximation

We consider the estimation of drift, diffusion, and noise covariance from discrete observations of stochastic differential equations driven by Gaussian processes. For a fixed observation horizon

T>0

and a known initial state

x_0\in\mathbb R

, we study \begin{equation*} dX_t=a(X_t),dt+σ(X_t),dZ_t^{β,f}, \qquad X_0=x_0,\quad 0\leq t\leq T. \end{equation*} \smallskip\noindent Here

a:\mathbb R\to\mathbb R

is the drift coefficient,

σ:\mathbb R\to(0,\infty)

is the diffusion coefficient, and

Z^{β,f}

is a centered Gaussian process from the weighted sub-fractional Brownian family, with covariance \begin{equation*} \operatorname{Cov}(Z_s^{β,f},Z_t^{β,f}) =\int_0^{s\wedge t} f(r)q_β(s-r,t-r),dr, \qquad 0\leq s,t\leq T. \end{equation*} \smallskip\noindent Here

s\wedge t=\min\{s,t\}

. The temporal weight

f:[0,T]\to[0,\infty)

is measurable, bounded, and positive almost everywhere, and

β\in(0,2)

is the covariance exponent. For

u,v\geq0

, the kernel is

q_β(u,v)=[u^β+v^β-(u+v)^β]/(1-β)

when

β\ne1

. Its continuous extension at

β=1

is

q_1(u,v)=(u+v)\log(u+v)-u\log u-v\log v

, with

0\log0=0

. Using the Euler approximation, we reconstruct the Gaussian driving increments from observed transitions and use their joint density to obtain a trajectory likelihood. Neural and radial-basis representations model the drift, diffusion, and normalized temporal weight, while a likelihood profile estimates the covariance exponent and diffusion scale. We compare the method with two neural alternatives on the same simulated trajectories in twenty coefficient settings.

J. H. Ramirez-Gonzalez

Institute of Mathematics and Statistics, University of São Paulo (IME–USP), São Paulo, Brazil

Parameter Estimation Stochastic Differential Equations Covariance Estimation

Sep 30, 2026eess.SP

Mean Spatial Frequency Decoupling for Learning-Based Uplink-to-Downlink Covariance Conversion in FDD Massive MIMO

In frequency division duplexing (FDD) massive multiple-input multiple-output (MIMO) systems, the uplink (UL)-to-downlink (DL) channel covariance matrix (CCM) conversion problem is studied to relieve the heavy burden of DL training and feedback required for channel estimation. Learning- based methods perform well up to a certain array size, but for a fixed dataset size their accuracy deteriorates with the number of antennas, to the point where simple model-based methods outperform them. This paper identifies a key cause of this behavior and addresses it. The mean angle of arrival (AoA) induces a phase ramp along the lags of the CCM. Since the oscillation rate of this ramp grows with the number of antennas, a dataset of fixed size becomes increasingly sparse relative to the variation that must be captured. We propose estimating the slope of this ramp from the UL CCM separately and mapping it to the DL band in closed form, leaving the learner with a residual that is largely insensitive to the mean AoA, which substantially reduces the performance degradation with an increasing number of antennas. The proposed scheme, termed deramping, is a combination of pre- and post-processing steps that applies to learning-based conversion methods without altering their internal structure, as demonstrated on three structurally different learners. Simulation results show that deramping reduces the covariance estimation error of all three learners under uniform, Laplacian, and Gaussian angular power spectra,keeps the interpolation-based learners ahead of a model-based benchmark at large array sizes, and improves downlink channel estimation.

Melih Can Zerin

Department of Electrical and Electronics Engineering Middle East Technical University Ankara, Türkiye

Channel Estimation Covariance Estimation

Sep 21, 2026cs.LG

Spatiotemporal Kronecker Covariance Neural Networks

Multivariate time series contain complex patterns that span across both space and time. While covariance-based statistical tools like spatiotemporal Principal Component Analysis (ST-PCA) help identify these patterns, they are limited to linear operations and prone to estimation errors with limited data. Recent covariance-based spatiotemporal neural networks offer more stable, non-linear alternatives, but they ignore correlations across different time steps. To solve this, we introduce the Kronecker coVariance Neural Network (KVNN), a temporal graph neural network that represents the spatiotemporal covariance matrix via a sum of Kronecker products where spatial and temporal dependencies are decoupled. By implementing filtering operations on spatial and temporal components, KVNNs achieve expressive processing capabilities, admit a rigorous spectral analysis, and are provably stable to finite-sample estimation errors, ultimately addressing all of ST-PCA's limitations. We show on five real-world datasets that KVNNs achieve strong forecasting performance, often requiring significantly fewer trainable parameters than competitive methods, and are consistent under estimation noise.

Andrea Cavallo, Athanasios Georgoutsos, Elvin Isufi

Delft University of Technology, Delft, Netherlands

Multivariate Time Series Forecasting Temporal GNNs Spatiotemporal Forecasting Covariance Estimation

Sep 18, 2026stat.ML

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.

Xiaotian Chang, Yangdi Jiang, Qirui Hu

School of Physical and Mathematical Sciences, Nanyang Technological University · School of Statistics and Data Science, Shanghai University of Finance and Economics.

Riemannian Optimization Confidence Region Estimation Stochastic Approximation Differential Privacy Covariance Estimation

Sep 8, 2026stat.ME

A Closed-Form Estimator and Diagnostic Battery for Anchor-Judge Error Correlation, Under a Single-Common-Factor Model

When an external reference set (an anchor) is used to decompose an LLM-judge panel's error into a quality signal and a shared common-mode error, standard practice assumes the anchor is uncontaminated: its error uncorrelated with the judges' shared error. We study when that assumption can be dropped and replaced by an estimate. Under a single-common-factor model, >=2 judges and >=2 anchors point-identify the quality variance, the common-mode variance, and each anchor's contamination correlation rho_k in closed form, with an exact per-anchor-pair failure boundary; a designated clean-anchor estimator, by contrast, reports a contaminated companion anchor as fully clean once its trusted anchor is itself contaminated. Because the single-common-factor assumption is itself untestable, the estimator ships gated behind a calibrated diagnostic battery (judge-covariance dispersion; over-identification; a family-block test from judge metadata, with a family-blocked estimator that removes family-level shared-residual bias exactly), bootstrap confidence intervals with measured coverage, and a weak-identification screen. A proposition maps which violations bias rho_k, in which direction, and which evade detection. For ordinal scores we show an identification hierarchy: with all variables ordinal, rho_k is not identified at any number of anchors; with ordinal judges and >=3 continuous anchors it is, and we give an estimator for that case. On real data the validation is asymmetric, and we say so plainly: the diagnostics are validated in the rejecting direction (both real panels we test are correctly rejected by the model-adequacy pre-test), while the estimator is validated in simulation and stress-tested semi-synthetically under oracle calibration; no real panel has yet passed the pre-test, and the pre-test exists precisely to say so. All results replay offline from shipped, checksummed artifacts.

Veerendra Kumar Sunkavalli

Independent Researcher

Latent Variable Models LLM-as-a-Judge Covariance Estimation

Aug 31, 2026q-fin.PM

End-to-End Neural Shrinkage of Indefinite Pairwise Correlation Matrices for Small-Cap-Inclusive Portfolios

Small-cap-inclusive equity universes contain recently listed and intermittently traded securities, so enforcing a common look-back discards a substantial fraction of the available information. Pairwise-complete estimation preserves the longest overlap for each asset pair, but the resulting correlation matrix can be indefinite because its entries are computed on different samples. This prevents direct use in Markowitz optimization and falls outside the assumptions of standard random-matrix shrinkage. We adapt a rotation-invariant neural covariance estimator to this setting. The model computes mask-aware marginal moments and a pairwise correlation matrix proxy, processes its signed spectrum, and uses a bidirectional gated recurrent unit conditioned on factor-aligned effective sample lengths derived from the overlap matrix and eigenvector loadings. It maps all eigenvalues, including negative ones, to a positive inverse spectrum. The reconstructed covariance is positive definite and is trained end-to-end to minimize five-day realized global-minimum-variance risk. We evaluate 26 expanding-window models from 2000 to 2025 on up to 1,500 U.S. equities in a closing-auction simulator with point-in-time selection, commissions, financing, corporate actions, and market impact. Across the 26-year out-of-sample period, the neural estimator reduces annualized five-day volatility by approximately 20% and increases the Sharpe ratio by approximately 40% relative to the next-best covariance estimator. These improvements are consistent across realized risk, risk-adjusted performance, and drawdown control, remain after the modeled execution frictions, and are supported by a 99.9% Model Confidence Set that retains only the neural estimator.

Christian Bongiorno, Lorenzo Villassero

Université Paris-Saclay, CentraleSupélec · Mathématiques et Informatique pour la Complexité et les Systèmes · Gif-sur-Yvette, France +2

Quantitative Finance Portfolio Optimization Covariance Estimation

Aug 4, 2026cs.LG

Unscented KalmanNet: Structure-Preserving Deep Learning with Calibrated Posterior Uncertainty under Incomplete Physics and Unknown Noise

Nonlinear state estimation requires sequentially fusing model-based predictions with noisy measurements. Under imperfect dynamics and unknown, time-varying noise statistics, this fusion can degrade in both accuracy and statistical consistency. Existing learning-aided filters largely treat accuracy and uncertainty estimation separately, limiting their ability to correct model-mismatch-induced bias while retaining an explicit, calibrated posterior covariance. This paper introduces Unscented KalmanNet (UKN), a model-based deep learning architecture that extends the Unscented Kalman Filter (UKF) with learned mechanisms for these two sources of filtering error while preserving explicit posterior covariance propagation. NoiseNet learns time-varying process and measurement covariances as bounded multiplicative corrections to baseline covariances, guaranteeing positive definiteness, while GainNet learns a bounded residual correction to the analytical UKF gain to compensate for model-mismatch-induced bias. A calibration-aware training objective couples state error with posterior covariance and innovation consistency terms through adaptive weighting, jointly optimizing accuracy and calibration. UKN is benchmarked against UKF, KalmanNet, and Bayesian KalmanNet on three synthetic systems and real-flight UZH-FPV data. It achieves the lowest state-estimation error in all four examples and reduces RMSE by 26.4-49.7% compared with UKF in the synthetic cases. Leave-one-sequence-out cross-validation over 11 flights shows 22.4% and 34.3% reductions in mean position and velocity RMSE, respectively. UKN also yields the lowest fold-to-fold variability, with dimension-normalized NEES and empirical coverage closest to nominal values among covariance-reporting filters. These results show that structured learned adaptation improves estimation accuracy while retaining calibrated uncertainty.

Minhyeok Ko, Abdollah Shafieezadeh

Department of Civil and Construction Engineering and Management, The University of Texas at Tyler, Tyler, TX, 75799, USA · Department of Civil, Environmental, and Geodetic Engineering, The Ohio State University, Columbus, OH, 43210, USA

Kalman Filtering Unscented Kalman Filtering Covariance Estimation Robot State Estimation Uncertainty Calibration

Jul 21, 2026stat.ME

Deep Shape Regression for Planar Curves with Multimodal Covariates

The shape of a planar curve is the geometric information that remains once translation, rotation, scale and reparametrisation are removed and is of interest in many health applications, e.g. in neuroimaging. We propose a deep shape regression model for open planar curves that admits multimodal and high-dimensional covariates. Representing curves as complex-valued functions, we show that the conditional full Procrustes mean is the leading eigenfunction of the conditional covariance. To estimate this covariance surface, we propose a novel deep conditional covariance smoother with modality-specific encoders - e.g. splines for scalar covariates and convolutional networks for images, which classical spline smoothers cannot accommodate. Our model is by construction invariant to the translation, rotation and scaling of the input curves and handles sparsely and irregularly sampled curves. We further provide an algorithm for elastic mean estimation that also removes parametrisation by iterating covariance smoothing, rotational alignment and parametrisation alignment. We illustrate the method on simulated outlines with known conditional mean and multimodal covariates, and give a first application to hippocampal outlines from the ADNI cohort, recovering covariate effects consistent with the literature. Code is available at https://github.com/mpff/dnn-shapes.

Manuel Pfeuffer, Roshan Prakash Rane, Hadya Yassin +2

Humboldt-Universität zu Berlin, Berlin, Germany · Hertie Institute for AI in Brain Health, Universität Tübingen, Tübingen, Germany · Universität Tübingen, Tübingen, Germany +1

Statistical Shape Modeling Covariance Estimation

Jul 9, 2026eess.SY

Adaptive MPPI with Online Disturbance Covariance Estimation: Provable Stability Tightening via Spatial Smoothing

We study Model Predictive Path Integral (MPPI) control for nonlinear systems with additive process disturbances whose covariance is unknown, spatially varying, and slowly time-varying. A mismatched disturbance covariance produces a persistent penalty in closed-loop stability certificates, while online estimation can reduce this penalty as data are collected. We propose a cell-wise recursive covariance estimator with spatial diffusion and prove a finite-horizon error bound that separates stochastic-approximation error, spatial-smoothing bias, and temporal-drift effects. The diffusion kernel is chosen to be reversible with respect to the stationary visitation measure, making the diffusion operator dissipative in the weighted Lyapunov analysis. We then substitute the resulting covariance estimate into the MPPI sampling distribution and derive an adaptive stability certificate with an explicit learning penalty. The main result is a payoff theorem: after a computable crossover time, the adaptive controller achieves a strictly tighter certified stability bound than any fixed covariance choice whose mismatch exceeds the residual smoothing and drift allowance. Numerical experiments illustrate the estimator convergence and the resulting stability-tightening effect.

Hyung-Jin Yoon, Hunmin Kim

Department of Mechanical and Nuclear Engineering, Tennessee Technological University, Cookeville, TN, USA

Lyapunov Stability Stochastic Approximation MPPI Control Covariance Estimation Adaptive Control

Jun 28, 2026cs.RO

Analyzing Uncertainty in the Spatial Representation of the Kinematic Bicycle Model

Locating a vehicle and determining its orientation in an uncertain environment is a critical challenge in autonomous vehicle navigation and path planning. To address these challenges, a vehicle estimates its pose while depending on sensor data that offer noisy measurements. These uncertainties in pose quantities are expressed mathematically as a covariance matrix. The real-time computation of the covariance matrix is critical because of the non-linearity involved in the kinematic model. The challenge is thus to evaluate the evolution of the covariance matrix of a vehicle's discretized stochastic kinematics. The purpose of this study is to obtain a near-accurate evolution of the covariance matrix of the rear-wheel bicycle kinematic model under uncertainties in wheel displacement and steering angle. We used Taylor's series to linearize the nonlinear trigonometric functions and provided closed-form expectations of random variables with the required accuracy. Our analytical findings are in good agreement with those obtained from Monte-Carlo simulations. Our contribution is probably the first detailed closed-form presentation of the covariance matrix constituents of the vehicle under evaluation, which were previously reported either incorrectly or incompletely. These findings aid in identifying the potential and constraints of the discretized kinematic model as well as its stochastic analysis. The techniques presented here are useful for the simultaneous localization and odometry self-calibration of certain mobile robots and autonomous vehicles.

Shafayat Abrar, M. Zaeem Baig, Shahir Ul Islam Anzal +1

Dhanani School of Science and Engineering Habib University, Karachi 75290

Uncertainty Quantification Covariance Estimation Robot State Estimation Robot Localization

Jun 25, 2026stat.ML

The Decision Geometry of Covariance Estimation for the Global Minimum-Variance Portfolio under Heavy Tails

The global minimum-variance portfolio (GMVP) is the canonical decision built from an estimated covariance matrix, yet covariance estimators are universally evaluated by matrix-norm loss, which is not the object the decision depends on. We characterise exactly how covariance-estimation error maps into GMVP suboptimality. We prove an exact regret identity and a non-asymptotic bound showing decision regret depends on the estimation error only through its action on the portfolio weights, scaled by portfolio concentration and the conditioning of the true covariance. From this we derive the decision geometry: GMVP regret is invariant to a (p-1)-dimensional projection of the p^2-dimensional error matrix, with invariance to the covariance-scale direction as an exact special case. We then apply the framework to heavy-tailed returns (tail index kappa in (2,4)), establishing the regret convergence rate implied by the centred operator-norm rate, and confirm the theory on a skew-t/t-copula simulation design with pre-registered analysis. The decision-focused advantage is a sharper constant and a concentration discount rather than a faster rate; we report an honest high-conditioning boundary of the rate prediction. The results complement recent decision-focused learning approaches by supplying the exact estimation geometry and consistency theory they lack.

Xavier Fonseca

Academy for AI, Games and Media, Breda University of Applied Sciences, Breda, The Netherlands.

Regret Minimization Portfolio Optimization Heavy-Tailed Distributions Covariance Estimation

Jun 22, 2026cs.LG

SOAP-Bubbles: Structured Weight Uncertainty for Neural Networks

Structured weight-uncertainty can improve many aspects of deep learning, but it remains costly to estimate and difficult to implement. Here, we show that these issues can be addressed by adapting the SOAP optimizer. Our key idea is to run IVON, an existing diagonal-covariance variational method, in the eigenspace of SOAP's preconditioner and then use the preconditioner to transform the diagonal estimate into a non-diagonal covariance. The resulting method has costs similar to those of SOAP and requires no drastic changes to training pipelines. We call the posteriors obtained in this way SOAP-Bubbles and our new optimizer Eigenspace-VON (EVON). We show that, for logistic regression, EVON recovers the exact Gaussian covariance and that, for language model pretraining, it yields significantly better results than existing diagonal-covariance methods. Our work makes it easier to estimate more expressive posterior distributions for deep learning at scale.

Adrian Robert Minut, Nico Daheim, Marco Miani +3

Sapienza University of Rome, Rome, Italy · Ubiquitous Knowledge Processing Lab (UKP Lab), Department of Computer Science, Technical University of Darmstadt · Technical University of Denmark, Lyngby, Denmark +3

Bayesian Neural Networks Neural Posterior Estimation Neural Network Optimization Variational Inference Covariance Estimation

Jun 20, 2026cs.LG

On the Curse of Dimensionality in Private Sparse Covariance Estimation and PCA

We study high-dimensional differentially private (DP) covariance estimation in the operator norm, and principal component analysis (PCA), under

k

-row-column sparsity (

k

-RCS) of the covariance matrix. In the non-private setting, it is known that

\mathsf{poly}(k, \log d)

samples suffice to solve both of these problems. However, the only comparable result known under DP (Wang et al. 2021) requires

Ω(d)

samples under standard parameterizations of the problem. We investigate when this curse of dimensionality is inherent for sparse covariance estimation tasks under DP. On the upper bound front, we show that a

\mathsf{poly}(k, \log d)

sample complexity for PCA is possible under DP, if we also posit sparsity of the leading eigenvector. We complement this result with

\mathsf{poly}(d)

lower bounds under DP for both sparse covariance estimation and PCA, establishing an exponential gap between the private and non-private variants of these problems when

k = \mathsf{polylog}(d)

. To our knowledge, no such separation has previously been demonstrated for any sparse estimation problems in private high-dimensional statistics. Our techniques are flexible enough that they imply stronger lower bounds even for the well-studied problem of standard DP PCA, without sparsity assumptions.

Syamantak Kumar, Shourya Pandey, Purnamrita Sarkar +1

University of Texas at Austin

Principal Component Analysis Minimax Estimation Differential Privacy Covariance Estimation

Jun 18, 2026cs.RO

ARC: Adaptive Robust Joint State and Covariance Estimation

Sensor measurements are frequently corrupted by outliers and non-Gaussian noise. These imperfections in the sensor data can cause classical state estimators to generate biased and unreliable state and uncertainty estimates. Robust estimators reject or downweight outliers but do not perform measurement covariance estimation, whereas joint state and covariance estimators assume Gaussian residuals and fixed loss shape parameters. Integrating these two capabilities into a single framework is an opportunity to simultaneously estimate both state and covariance in the presence of outliers. This paper proposes a unified Block-Coordinate Descent framework that combines a norm-aware adaptive robust loss, an Iteratively Reweighted Least-Squares state update, and a Minimum Weighted Covariance Determinant covariance estimator, yielding a self-tuning joint state and covariance estimator. The framework is evaluated in a Monte-Carlo simulation and on real-world ultra-wideband localization experiments in cluttered non-line-of-sight environments. Results show that the proposed estimator consistently recovers the true inlier measurement covariance and matches or exceeds the state estimation accuracy of all baselines, without requiring any manual parameter tuning.

Alexandre Hadji-Thomas, Andrew Stirling, James R. Forbes

Covariance Estimation Robot State Estimation Robust Estimation

Jun 14, 2026cs.LG

Ablation, Statistical Inference, and Validation for KV-Cache Compression

This study systematically compares Turbo-Quant and SpectralQuant KV-cache compression, evaluating non-dominated schemes, including WHT rotation with Beta Lloyd-Max and QJL, through a statistical validation methodology that separates systematic codec differences from implementation variance. Key findings reveal that while eigenbasis-based methods fail on heavy-tailed data due to covariance instability, they excel in structured regimes, with the effective semantic dimension (

d_{eff}

) adapting to calibration budgets rather than true data rank. (this is an abstract of the abstract thank you )

Paolo D'Alberto, Ashish Siarasao, Elliott Delaye +1

Advanced Micro Devices, Inc.

KV Caching Rotation-Based Quantization KV-Cache Compression Covariance Estimation KV-Cache Quantization Statistical Hypothesis Testing

Jun 5, 2026cs.LG

Covariance Shrinkage via Stochastic Interpolation

We recast classical shrinkage of high-dimensional covariance estimators as empirical risk minimization over a parametric stochastic interpolant between a source and a target distribution. This formalism recovers known shrinkage estimators as special cases and reveals three distinct mechanisms for reducing statistical risk: (i) Scheduling: the interpolant schedule determines the class of admissible covariances, and hence the achievable risk. (ii) Flow maps and couplings: whereas naive constructions amount to assuming independence between the distributions, specific coupling structures (e.g., solutions of optimal transport problems) can lower the empirical risk. Moreover, non-linear flow maps realizing such couplings free the interpolant covariance from the eigenbasis of the empirical estimate, enabling eigenvector regularization. (iii) Early stopping: estimators defined by integrating a regressed vector field afford an additional bias-variance trade-off through approximation of the true interpolant distribution. We then propose a neural estimator of the interpolant, together with an upper bound on its quadratic risk in terms of the interpolant approximation error, and validate both on synthetic experiments. Finally, we apply the estimator to real neuroimaging data, demonstrating the additional regularization power this approach offers in practice.

Mathieu Chalvidal, Florentin Coeurdoux, Eric Vanden-Eijnden

Capital Fund Management 23 Rue de l’Universit´e, 75007 Paris

Stochastic Interpolants Covariance Estimation

May 26, 2026stat.ML

Semiparametrically Efficient Inference for Kernel Measures of Noise Heterogeneity

We develop semiparametrically efficient inference for kernel measures of noise heterogeneity in additive noise models. In many applications, the regression function is estimated using flexible machine learning methods. Downstream procedures based on the resulting residuals can then inherit first-stage bias: regression error may induce spurious dependence between covariates and residuals, invalidating the assumptions needed for standard analysis. We construct a novel Hilbert-valued one-step estimator of the kernel covariance operator between covariates and residuals. Our estimator yields bootstrap-calibrated tests for residual independence and goodness of fit in additive noise models, while also providing asymptotically efficient confidence intervals for the kernel dependence measure under noise heterogeneity. The framework extends to settings with additional covariates, enabling inference on distributional heterogeneity of residual noise across treatment groups. Simulations show improved calibration and power relative to naive plug-in residual methods.

Jakub Wornbard, Zikai Shen, Dimitri Meunier +1

University College London

Semiparametric Inference Kernel Methods Covariance Estimation Statistical Hypothesis Testing

May 26, 2026stat.ML

Gaussian Process-based learning with new MCMC-based implementation of Wishart prior on correlation matrix

In probabilstic supervised learning of an input-output relationship - as a sample function of a Gaussian Process (GP) - priors are typically specified for the hyperparameters of the kernel that parametrises the covariance function of the GP, where the induced covariance matrix of the (resulting multivariate Normal) likelihood, governs the learning and prediction. When the sought function is highly multivariate, multiple lengthscale parameters must be learnt simultaneously, making inference difficult. We develop a ``self-assembled'' Wishart prior for the covariance matrix, while undertaking Bayesian inference on the kernel hyperparameters using MCMC. The construction uses a look-back window over recent MCMC iterations to define a time-step dependent scale matrix, thereby introducing adaptiveness to the chain. Results suggest that direct prior specification on the covariance matrix can be useful for diagnosing weakly informative inputs within the GP-based learning paradigm. We support our prior development with two distinct empirical illustrations - one on synthetic data, and another on a real-world dataset.

Kane Warrior, Dalia Chakrabarty

Department of Mathematics University of York

Bayesian Inference Covariance Estimation Gaussian Process Regression

May 22, 2026cs.LG

Private Adaptive Covariance Estimation via Gaussian Graphical Models

We propose PACE-GGM, a data-adaptive differentially private method for covariance estimation that concentrates its privacy budget on the most informative entries of the empirical covariance matrix, rather than perturbing all entries. This applies in the natural setting where the modeler supplies separate bounds for each variable, so that individual entries can be measured with less noise than the full matrix. In each round, our method selects a poorly approximated entry, measures it using the Gaussian mechanism, and then reconstructs a full covariance matrix using a maximum-entropy reconstruction objective, leading to a Gaussian graphical model structure. Experiments on diverse real-world datasets demonstrate consistent improvements in estimation error with respect to the Gaussian mechanism and other baselines, particularly in high-dimensional and low-to-moderate privacy regimes.

Cecilia Ferrando, Miguel Fuentes, Brett Mullins +2

Manning College of Information and Computer Sciences University of Massachusetts Amherst

Differential Privacy Covariance Estimation

May 18, 2026math.ST

Self-Distillation is Optimal Among Spectral Shrinkage Estimators in Spiked Covariance Models

Self-distillation has emerged as a promising technique for improving model performance in modern machine learning systems. We develop the statistical foundations of self-distillation in spiked covariance models, by introducing and analyzing a broad class of estimators, namely spectral shrinkage estimators. We establish that for spiked covariance matrices with

s

spikes,

s

-step self-distillation achieves optimal performance among spectral shrinkage estimators, outperforming well-known estimators in statistics and machine learning. Moreover, we show that

s

steps are necessary for optimality: any

(s-k)

-step distilled estimator is strictly suboptimal for

1 \leq k \leq s

. For the special subclass of isotropic covariances, we show that optimally tuned Ridge regression performs best among spectral shrinkage estimators. We also study a federated approach where multiple data centers share spectral shrinkage estimators and a common server seeks to aggregate them to achieve optimal performance. In this case, we find that the best local rule again takes the form of self-distillation, though it differs from the optimal rule when data are hosted centrally on a single server. Together, our results elucidate why self-distillation improves predictive performance and provide a broader statistical framework connecting it with classical shrinkage-based methods.

Radu Lecoiu, Debarghya Mukherjee, Pragya Sur

Department of Statistics, Harvard University · Department of Mathematics & Statistics, Boston University

Self-Distillation Spectral Methods Covariance Estimation Statistical Learning Theory

May 17, 2026q-fin.RM

A Hybrid Gaussian Process Regression Framework for Stable Volatility-Covariance Estimation: Evidence from Global Equity Indices

Accurate forecasting of the Volatility-Covariance Matrix (VCV) is central to regulatory capital adequacy processes such as the Internal Capital Adequacy Assessment Process (ICAAP) and the Comprehensive Capital Analysis and Review (CCAR). Traditional econometric models, including GARCH-family and Exponentially Weighted Moving Average (EWMA) approaches, suffer from parametric rigidity, distributional assumptions, and numerical instability under stress, leading to systematic underestimation of tail risk. This paper proposes and validates a novel Hybrid Gaussian Process Regression-Historical Simulation (GPR-HS) framework for estimating Value-at-Risk (VaR) and Expected Shortfall (ES) across a diversified portfolio of seven major global equity indices. The framework decouples the VCV estimation problem: individual asset volatilities are modelled dynamically using Univariate GPR with a Matern 5/2 kernel, while inter-asset correlations are estimated via stable historical covariance. A key methodological contribution is the Aggressive Noise Initialization (ANI) strategy, which sets the initial White Noise kernel variance equal to the empirical variance of the training returns, ensuring Gram matrix positive-definiteness, regularization, and conservative, regulatory-compliant forecasts. Evaluated using an expanding window forward-chaining cross-validation scheme over June 2020 -June 2025, the GPR-HS framework achieves regulatory compliance in the majority of test splits; including a 100% ES pass rate at the portfolio level, while outperforming the static Historical VaR benchmark in 71.4% of univariate cases by Quadratic Loss and 100% of cases by violation count.

Ujjwala Vadrevu

Quantitative Finance Financial Time Series Covariance Estimation Gaussian Process Regression

May 13, 2026stat.ML

Covariance-aware sampling for Diffusion Models

We present a covariance-aware sampler that improves the quality of pixel-space Diffusion Model (DM) sampling in the few-step regime. We hypothesize that in the few-step regime samplers fail because they rely solely on the predicted mean of the reverse distribution, while our solution explicitly models the reverse-process covariance. Our method combines Tweedie's formula to estimate the covariance with an efficient, structured Fourier-space decomposition of the covariance matrix. Implemented as an extension of DDIM, our method requires only a minimal overhead: one extra Jacobian-Vector Product (JVP) per step. We demonstrate that for pixel-based DMs, our method consistently produces superior samples compared to state-of-the-art second order samplers (Heun, DPM-Solver++) and the recent aDDIM sampler, at an identical number of function evaluations (NFE).

Andrea Schioppa, Tim Salimans

GDM - Amsterdam · work done at GDM - Amsterdam

Diffusion Model Sampling Diffusion Sampling Pixel-Space Diffusion Models Few-Step Diffusion Sampling Covariance Estimation

May 8, 2026cs.SD

Online Segmented Beamforming via Dynamic Programming

In dynamic acoustic environments characterized by time-varying interferers and moving sources, effective beamforming requires accurately identifying stationary regions over time. Traditional Capon beamformers rely on the instantaneous ensemble covariance matrix, which is inaccessible in practice. Practical implementations overcome this by estimating the sample covariance matrix (SCM) through averaging over a block of temporal samples. However, in non-stationary settings, a naive batch approach fails. Moving interferers smear the SCM, causing the beamformer to place nulls in outdated locations while failing to track newly active interferers, thereby degrading its nulling capabilities. To address this fundamental limitation, an Online Segmented Beamformer is proposed. This algorithm incorporates data-driven temporal segmentation to causally minimize output power while dynamically adapting the SCM estimation windows to local stationarity. By framing the problem through the lens of dynamic programming, the proposed method tracks abrupt environmental changes and resets covariance estimates in real-time. We validate the performance of this framework in a complex, reverberant simulated acoustic environment and in highly reverberant real world experiments, demonstrating its superiority over fixed-window adaptive methods.

Manan Mittal, Ryan M. Corey, Diego Cuji +2

Stony Brook University1 · University of Illinois Chicago2 · University of Massachusetts Darthmouth3

Dynamic Programming Covariance Estimation

May 1, 2026cs.LG

Divergence is Uncertainty: A Closed-Form Posterior Covariance for Flow Matching

Flow matching has become a leading framework for generative modeling, but quantifying the uncertainty of its samples remains an open problem. Existing approaches retrain the model with auxiliary variance heads, maintain costly ensembles, or propagate approximate covariance through many integration steps, trading off training cost, inference cost, or accuracy. We show that none of these trade-offs is necessary. By extending Tweedie's formula from the denoising setting to the flow matching interpolant, we derive an exact, closed-form expression for the posterior covariance at every point along the generative trajectory. The result depends on a single quantity, namely the divergence of the learned velocity field, which can be computed post-hoc on any pre-trained flow matching model, requiring no retraining and no architectural modification. For one-step generators such as MeanFlow, the same formula yields the end-to-end generation uncertainty in a single forward pass, eliminating the multi-step variance propagation required by all prior methods. Experiments on MNIST confirm that the resulting per-pixel uncertainty maps are semantically meaningful, concentrating on digit boundaries where inter-sample variation is highest, and that the scalar uncertainty score tracks actual prediction error, all at roughly

10^4 \times

less total compute than ensembling or Monte Carlo dropout.

Jiarui Xing, Song Wang, Jian Wang

Yale University · Harvard Medical School

Flow Matching Uncertainty Quantification Bayesian Inference Flow-Based Generative Modeling Covariance Estimation Uncertainty Propagation

Apr 25, 2026stat.ML

Inference of Online Newton Methods with Nesterov's Accelerated Sketching

Reliable decision-making with streaming data requires principled uncertainty quantification of online methods. While first-order methods enable efficient iterate updates, their inference procedures still require updating proper (covariance) matrices, incurring

O(d^2)

time and memory complexity, and are sensitive to ill-conditioning and noise heterogeneity of the problem. This costly inference task offers an opportunity for more robust second-order methods, which are, however, bottlenecked by solving Newton systems with

O(d^3)

complexity. In this paper, we address this gap by studying an online Newton method with Hessian averaging, where the Newton direction at each step is approximately computed using a sketch-and-project solver with Nesterov's acceleration, matching

O(d^2)

complexity of first-order methods. For the proposed method, we quantify its uncertainty arising from both random data and randomized computation. Under standard smoothness and moment conditions, we establish global almost-sure convergence, prove asymptotic normality of the last iterate with a limiting covariance characterized by a Lyapunov equation, and develop a fully online covariance estimator with non-asymptotic convergence guarantees. We also connect the resulting uncertainty quantification to that of exact and sketched Newton methods without Nesterov's acceleration. Extensive experiments on regression models demonstrate the superiority of the proposed method for online inference.

Haoxuan Wang, Xinchen Du, Sen Na

School of Industrial and Systems Engineering, Georgia Institute of Technology.

Stochastic Approximation Second-Order Optimization Randomized Sketching Covariance Estimation

Apr 23, 2026stat.ML

Refining Covariance Matrix Estimation in Stochastic Gradient Descent Through Bias Reduction

We study online inference and asymptotic covariance estimation for the stochastic gradient descent (SGD) algorithm. While classical methods (such as plug-in and batch-means estimators) are available, they either require inaccessible second-order (Hessian) information or suffer from slow convergence. To address these challenges, we propose a novel, fully online de-biased covariance estimator that eliminates the need for second-order derivatives while significantly improving estimation accuracy. Our method employs a bias-reduction technique to achieve a convergence rate of

n^{(α-1)/2} \sqrt{\log n}

, outperforming existing Hessian-free alternatives.

Ziyang Wei, Wanrong Zhu, Jingyang Lyu +1

Department of Statistics University of Chicago Chicago, IL 60637, USA · Department of Statistics University of California, Irvine Irvine, CA 92617, USA · Department of Statistics University of Wisconsin–Madison Madison, WI 53706, USA

Covariance Estimation Stochastic Gradient Descent

Apr 1, 2026cs.LG

Model Merging via Data-Free Covariance Estimation

Model merging provides a way of cheaply combining individual models to produce a model that inherits each individual's capabilities. While some merging methods can approach the performance of multitask training, they are often heuristically motivated and lack theoretical justification. A principled alternative is to pose model merging as a layer-wise optimization problem that directly minimizes interference between tasks. However, this formulation requires estimating per-layer covariance matrices from data, which may not be available when performing merging. In contrast, many of the heuristically-motivated methods do not require auxiliary data, making them practically advantageous. In this work, we revisit the interference minimization framework and show that, under certain conditions, covariance matrices can be estimated directly from difference matrices, eliminating the need for data while also reducing computational costs. We validate our approach across vision and language benchmarks on models ranging from 86M parameters to 7B parameters, outperforming previous data-free state-of-the-art merging methods

Marawan Gamal Abdel Hameed, Derek Tam, Pascal Jr Tikeng Notsawo +2

Mila & DIRO, Universit´e de Montr´eal · University of Toronto & Vector Institute

Multi-Task Learning Language Model Merging Model Merging Covariance Estimation

Nov 3, 2025cs.LG

The Optimization Landscape of Carathéodory Decomposition of Toeplitz Covariances

Toeplitz covariance estimation is a classical problem in statistical signal processing, yet the geometry of the Gaussian maximum-likelihood objective remains only partially understood. Recent algorithms, including Newton-type, majorization-minimization, and gradient-based methods, indicate that the nonconvex problem can often be globally solved when the number of samples is sufficiently large, but they also reveal a difficult computational landscape. In this work, we study this phenomenon through an overparameterized Caratheodory representation of positive definite Toeplitz covariance matrices. The Caratheodory decomposition parameterizes the covariance using a combination of steering vectors with different frequencies and amplitudes. Our first result shows that fixed-grid amplitude optimization is fundamentally insufficient. Even in the population setting, and even with arbitrarily many fixed frequency grid points, amplitude-only optimization can have a strictly positive error floor under grid mismatch. This motivates optimizing both amplitudes and frequencies. In this case, our main theoretical result proves that the joint optimization has a benign population landscape: every stationary point that produces a positive definite covariance matrix recovers the true Toeplitz covariance. These findings suggest a simple interpretation of the Toeplitz covariance problem: the population landscape is globally benign, but may be highly ill-conditioned. In our numerical experiments, overparameterization improves convergence speed and finite-sample accuracy. In particular, it allows simple gradient descent to approach the Cramer Rao bound while keeping the implementation simple.

Daniel Busbib, Ami Wiesel

School of Computer Science and Engineering, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Nonconvex Optimization Matrix Optimization Covariance Estimation

Oct 21, 2024stat.ML

Statistical Inference for Policy Evaluation with Temporal Difference Learning

We investigate the statistical properties of Temporal Difference (TD) learning with Polyak-Ruppert averaging, arguably one of the most widely used algorithms in reinforcement learning, for the task of estimating the parameters of the optimal linear approximation to the value function. Assuming independent samples, we make three theoretical contributions that improve upon the current state-of-the-art results: (i) we establish refined high-dimensional Berry-Esseen bounds over the class of convex sets, achieving faster rates than the best known results, and (ii) we propose and analyze a novel, computationally efficient online plug-in estimator of the asymptotic covariance matrix; (iii) we derive sharper high probability convergence guarantees that depend explicitly on the asymptotic variance and hold under weaker conditions than those adopted in the literature. These results enable the construction of confidence regions and simultaneous confidence intervals for the linear parameters of the value function approximation, with guaranteed finite-sample coverage. We demonstrate the applicability of our theoretical findings through numerical experiments.

Weichen Wu, Gen Li, Yuting Wei +1

The Voleon Group, New York, NY, USA · Department of Statistics, The Chinese University of Hong Kong, Hong Kong SAR. · Department of Statistics and Data Science, The Wharton School, University of Pennsylvania, Philadelphia, PA, USA. +1

Parameter Estimation Confidence Region Estimation Value Function Estimation Stochastic Approximation Policy Evaluation Temporal-Difference Learning+1