Optimal first-order methods are shown to be fundamental limits in functional estimation.
problem Optimal functional estimation under weak conditions.
method Formalization of functional estimation with black-box nuisance function estimates and derivation of minimax lower bounds.
result First-order methods are optimal under weak conditions, but higher-order methods can outperform them when nuisance function structure is known.
New estimator stabilizes higher-order influence functions for stable statistical inference.
problem Numerical instability in estimating inverse population Gram matrix.
method Proposes a new stabilized higher-order estimator without sample splitting.
result Stabilized estimator exhibits more stable performance and similar statistical guarantees.
New estimator stabilizes higher-order influence functions for bilinear forms.
problem Stability issues in estimating bilinear forms using higher-order influence functions.
method Proposes a new stabilized higher-order estimator for a class of bilinear forms without sample splitting.
result New estimator exhibits more stable finite-sample performance compared to the empirical higher-order estimator.
Simplified argument for second order estimate in quaternionic Calabi-Yau problem.
problem Second order estimates for quaternionic Calabi-Yau problem on hyperkähler manifolds.
method Simplified argument to derive the estimate.
result Simplified derivation of second order estimate.
Paper estimates the order of vertices in random recursive trees.
problem Estimating the order of arrival of vertices in random recursive trees.
method Proposes an order estimator based on the Jordan centrality measure and defines risk measures.
result Establishes a nearly optimal estimator for the problem.
In this paper, we consider lower order eigenvalues of Laplacian operator with any order in Euclidean domains. By choosing special rectangular coordinates, we obtain two estimates for lower order eigenvalues.
Estimates multiple related causal graphs with shared causal order.
problem Discovering multiple related Gaussian DAGs with shared causal order.
method Proposes a l1/l2-regularized MLE for joint estimation of K linear structural equation models. result Joint estimator achieves better sample complexity and consistency in causal order recovery.
A method to estimate high order derivatives of data distributions from samples.
problem Estimating high order derivatives of data distributions efficiently and accurately.
method Generalizing denoising score matching via Tweedie's formula to estimate higher order derivatives.
result Models trained with the proposed method can approximate second order derivatives more efficiently and accurately than via automatic differentiation.
Study proves inequalities for eigenvalues of fourth-order elliptic operators on Riemannian manifolds.
problem Eigenvalue inequalities for fourth-order elliptic operators on Riemannian manifolds.
method Proves inequalities using Payne-Pólya-Weinberger-Yang type for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds.
result Generalizes eigenvalue inequalities for the clamped plate problem to complete Riemannian manifolds.
New Monte Carlo method outperforms existing strategy for estimating Sobol' indices.
problem Estimating first-and total-orders Sobol' indices accurately.
method Comparing two Monte Carlo estimators for Sobol' indices.
result New method outperforms current approach in accuracy.
Paper derives estimates for complex Hessian equations on Hermitian manifolds.
problem Estimating solutions to complex Hessian equations on Hermitian manifolds.
method Derives second order estimates for solutions in a specific cone.
result Establishes second order estimates for solutions in Γk+1 cone. New Hessian estimators for Riemannian manifolds with reduced bias.
problem Estimating Hessians on Riemannian manifolds with reduced bias and computational efficiency.
method Introducing new stochastic zeroth-order Hessian estimators using O(1) function evaluations. result Achieved a bias bound of order O(γδ2) for analytic real-valued functions. Paper proposes methods to learn DAGs from partial orderings.
problem Learning DAGs from partial orderings is challenging.
method General estimation framework and efficient algorithms for low- and high-dimensional problems.
result Efficient estimation of DAGs from partial orderings is possible.
New protocols show 1-bit mean estimation can be order-optimal without interaction.
problem Can 1-bit mean estimation be optimal without interaction?
method Adaptive and non-adaptive threshold and interval queries, with one adaptive transition.
result Arbitrary non-adaptive quantizers can match the adaptive rate, suggesting interaction is not necessary.
The paper provides estimates for higher-order Ricci curvature along Kähler-Ricci flows.
problem Estimating higher-order curvature along Kähler-Ricci flows on compact Kähler manifolds.
method Proving uniform bounds for Ricci curvature and scalar curvature in various orders and norms.
result A geometric obstruction causes a specific third-order derivative of Ricci curvature to blow up at rate et/2. New method estimates mixture model components efficiently.
problem Estimating the number of components in finite mixture models.
method Group-Sort-Fuse (GSF) procedure for simultaneous estimation of order and mixing measure.
result GSF achieves consistent estimation of true mixture order and n−1/2 convergence rate. Enhances gradient estimates for Hermitian Monge-Ampère equations.
problem Improving estimates for Hermitian Monge-Ampère equations.
method Improves gradient estimates using Evans-Krylov and third derivatives estimates.
result Enhanced estimates for second and third order derivatives.
Paper proposes an online covariance estimator for sketched Newton methods.
problem Estimating the limiting covariance matrix of sketched Newton methods.
method Proposes a fully online covariance matrix estimator from Newton iterates.
result Establishes the consistency and convergence rate of the proposed estimator.
The study examines order flow in financial markets using fractional Lévy stable motion.
problem Challenges in selecting the best models for financial time series data.
method Investigates order disbalance time series from the perspective of fractional Lévy stable motion.
result Orders exhibit stable anti-correlation for 18 randomly selected stocks.
The paper analyzes the efficiency of gradient estimation methods in noisy function evaluations.
problem Estimating gradients of smooth functions using noisy function evaluations.
method Information-theoretic lower bounds and finite difference method analysis.
result The finite difference method is not minimax optimal, suggesting room for improvement in gradient estimation.
Improved estimation of higher order integrals using shrinkage techniques.
problem Estimating higher order Bochner integrals in non-parametric settings.
method Shrinkage of U-statistic towards a target element, considering kernel degeneracy.
result Consistent shrinkage estimators with fast rates of convergence, even for non-degenerate kernels.
Sharp spectral gap estimates for higher-order operators on hyperbolic spaces.
problem Estimating spectral gaps for higher-order operators on Cartan-Hadamard manifolds.
method Symmetrization-free proofs based on general functional inequalities.
result Solves a sharp asymptotic problem from Cheng and Yang and answers a question from Kristály.
For a bounded domain Ω in a complete Riemannian manifold Mn, we study estimates for lower order eigenvalues of a clamped plate problem. We obtain universal inequalities for lower order eigenvalues. We would like to remark that our results are sharp.
The modified Cholesky decomposition is commonly used for precision matrix estimation given a specified order of random variables. However, the order of variables is often not available or cannot be pre-determined. In this work, we propose to address the variable order issue in the modified Cholesky decomposition for sp…
We consider first order expansions of convex penalized estimators in high-dimensional regression problems with random designs. Our setting includes linear regression and logistic regression as special cases. For a given penalty function h and the corresponding penalized estimator β^, we construct a quantity η,…
Estimates hypergraphons for modeling complex interactions efficiently.
problem Modeling higher-order interactions using hypergraphons.
method Restricted class of Simple Lipschitz Hypergraphons (SLH) for efficient estimation.
result Optimal rates of convergence for SLH estimator.
Gradient-based methods for optimisation of objectives in stochastic settings with unknown or intractable dynamics require estimators of derivatives. We derive an objective that, under automatic differentiation, produces low-variance unbiased estimators of derivatives at any order. Our objective is compatible with arbit…
Networks are a natural representation of complex systems across the sciences, and higher-order dependencies are central to the understanding and modeling of these systems. However, in many practical applications such as online social networks, networks are massive, dynamic, and naturally streaming, where pairwise inter…
Study eigenvalues and eigenfunctions of fourth-order operators in annuli, proving optimal estimates and non-radiality.
problem Eigenvalue and eigenfunction analysis of fourth-order operators in degenerating annuli.
method Optimal estimates and non-radiality results for eigenfunctions in annuli.
result Nigh optimal estimate for the first eigenvalue and non-radiality of eigenfunctions in degenerating annuli.
We consider to learn a causal ordering of variables in a linear non-Gaussian acyclic model called LiNGAM. Several existing methods have been shown to consistently estimate a causal ordering assuming that all the model assumptions are correct. But, the estimation results could be distorted if some assumptions actually a…
The study provides interior estimates for Qk-flows and translators in Rn+1.
problem Estimating Qk-flows and translators in Rn+1. method Proved interior gradient and second order estimates.
result Non-existence of Qk-translators asymptotic to o(∣x∣). A Hawkes process with state-dependent factor models order flows in limit order books.
problem Modeling order flows in limit order books for better market prediction.
method A Hawkes process with a state-dependent factor for conditional intensity estimation.
result State-dependent formulations improve the fit of LOB models to financial data.
The paper introduces a frequency-domain estimator for low-order systems from noisy data.
problem Estimating frequency responses of low-order systems from noisy measurements.
method Uses a quadratic data-fitting term regularized by the nuclear norm of a Loewner matrix, subject to a convex stability constraint.
result Proves a finite-sample error bound and extends it to all frequencies through rational interpolation.
The paper tackles estimation of hidden state LTI systems of unknown order.
problem Estimation of Markov parameters and minimal realization of unknown order LTI systems.
method Hankel penalized least square estimator, Ho-Kalman algorithm, and a combined algorithm.
result Statistical guarantees for estimation error, rank recovery, and sample complexity.
New methods improve estimation accuracy in noisy settings.
problem Estimating treatment effects in the presence of treatment noise.
method Developed new structure-agnostic cumulant estimators and practical procedures for higher-order robustness.
result Demonstrated that existing DML estimator is suboptimal for non-Gaussian treatment noise and introduced ACE procedures for improved accuracy.
Paper studies Laplace operator estimates in harmonic map heat flows.
problem Estimating Laplace operator in harmonic map heat flows outside singularities.
method Investigates estimates using spherical coordinates for T2 and T3 boundary conditions. result Provides higher-order estimates for the Ericksen--Leslie system.
Researchers find second-order estimates for p-Laplacian in RCD spaces.
problem Estimating functions with p-Laplacian in RCD spaces. method Establishing quantitative second-order Sobolev regularity.
result Second-order estimates for p-Laplacian functions in RCD spaces. A new method predicts higher-order interactions in evolving graphs using simplicial complexes.
problem Predicting higher-order interactions in dynamic graphs with theoretical guarantees.
method Capturing higher-order interactions as simplices, modeling neighborhoods with face-vectors, and developing a nonparametric kernel estimator.
result Our method outperforms existing higher-order prediction methods and is theoretically consistent.
Regularization is a popular technique in machine learning for model estimation and avoiding overfitting. Prior studies have found that modern ordered regularization can be more effective in handling highly correlated, high-dimensional data than traditional regularization. The reason stems from the fact that the ordered…
Estimates for complex Hessian equations on Hermitian manifolds.
problem Establishing estimates for solutions to complex Hessian equations.
method Using concavity inequality for complex sum-of-Hessian operators.
result Second-order estimates for admissible solutions on Hermitian manifolds.
The purpose of this paper is to prove the a priori estimates for constant scalar curvature Kaehler metrics with conic singularities along normal crossing divisors. The zero order estimates are proved by a reformulated version of Alexandrov's maximum principle. The higher order estimates follow from Chen-Cheng's frame …
We obtain sharp estimates involving the mean curvatures of higher order of a complete bounded hypersurface immersed in a complete Riemannian manifold. Similar results are also given for complete spacelike hypersurfaces in Lorentzian ambient spaces.
Zeroth-order optimization methods lack inherent privacy guarantees.
problem Ensuring differential privacy in zeroth-order optimization methods.
method Analyzing ZO-GD with and without random initialization for convex and strongly convex objectives.
result ZO-GD is not differentially private for strongly convex objectives and can have superlinear privacy loss.
In this work, we develop an importance sampling estimator by coupling the reduced-order model and the generative model in a problem setting of uncertainty quantification. The target is to estimate the probability that the quantity of interest (QoI) in a complex system is beyond a given threshold. To avoid the prohibiti…
A new method for MARL with partial observations reduces communication overhead.
problem Inefficient MARL algorithms in large-scale problems due to state and action information sharing.
method Distributed zeroth-order policy optimization with local policy gradient estimation using consensus.
result The method converges to a policy that is a stationary point of the global objective function.
Study improves BN TTA under distribution shift using higher-order asymptotics.
problem Improving BN TTA for changing data distributions.
method Integrates Edgeworth expansion and saddlepoint approximation with one-step M-estimation.
result Derives optimal weighting parameter for minimized mean-squared error.
Estimates for harmonic functions in curved spaces.
problem Quantifying harmonic functions in curved spaces.
method Quantitative Sobolev estimates for p-harmonic functions in manifolds with curvature conditions. result Established a quantitative second order Sobolev estimate for p-harmonic functions. A new method learns DAGs from Gaussian data without verifying acyclicity.
problem Learning DAGs from Gaussian data without verifying acyclicity.
method Relaxation technique for permutation matrix estimation and cyclic coordinatewise descent for sparse Cholesky factor estimation.
result The method recovers DAGs without verifying acyclicity constraints.