Estimates eigenvalue for manifolds with specific forms under certain conditions.
problem Estimating the first eigenvalue of Laplacian on manifolds with almost parallel p-forms. method Uses Lichnerowicz-Obata estimate and pinching conditions to analyze eigenvalues.
result Establishes a Lichnerowicz-Obata type estimate for the first eigenvalue.
We prove the Finsler analog of the conformal Lichnerowicz-Obata conjecture showing that a complete and essential conformal vector field on a non-Riemannian Finsler manifold is a homothetic vector field of a Minkowski metric.
Proves rigidity for eigenvalue estimate on three-manifolds.
problem Eigenvalue estimate for Kohn Laplacian on three-manifolds.
method Rigidity proof for Lichnerowicz-type estimate.
result Rigidity for eigenvalue estimate on specific three-manifolds.
New eigenvalue bounds for 3-Sasaki metrics improve previous estimates.
problem Estimating eigenvalues for 3-Sasaki metrics.
method Improved Lichnerowicz-Obata type estimates for scalar sub-Laplacian.
result Lower bounds for the first non-zero eigenvalue of 3-Sasaki metrics.
The motivation of this paper is to study a second order elliptic operator which appears naturally in Riemannian geometry, for instance in the study of hypersurfaces with constant r-mean curvature. We prove a generalized Bochner-type formula for such a kind of operators and as applications we obtain some sharp estimat…
We discuss the behavior of (λ1.p(M))1/p with respect to the Gromov-Hausdorff topology and the variable p, where λ1,p(M) is the first positive eigenvalue of the p-Laplacian on a compact Riemannian manifold M. Applications include new estimates for the first eigenvalues of the p-Laplacian on Rieman…
We report on some aspects and recent progress in certain problems in the sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the corresponding Yamabe problems on the round spheres, the Lichnerowicz-Obata first eigenvalue estimates, and the relation between these two problems. A motivation from the…
We solve two classical conjectures by showing that if an action of a connected Lie group on a complete Riemannian manifold preserves the geodesics (considered as unparameterized curves), then the metric has constant positive sectional curvature, or the group acts by affine transformations.
We correct a mistake in Shen Yibing, Yu Yaoyong, On Projectively Related Randers Metrics, International Journal of Mathematics 19}(2008), no. 5, 503--520, and prove the natural generalization of the projective Lichnerowicz-Obata conjecture for Randers metrics.
We prove an inequality that generalizes the Fan-Taussky-Todd discrete analog of the Wirtinger inequality. It is equivalent to an estimate on the spectral gap of a weighted discrete Laplacian on the circle. The proof uses a geometric construction related to the discrete isoperimetric problem on the surface of a cone. In…
Inspired by the Lichnerowicz-Obata theorem for the first eigenvalue of the Laplacian, we define a new family of invariants {Ωk(g)} for closed Riemannian manifolds. The value of Ωk(g) delicately reflects the spherical part of the manifold. Indeed, Ω1(g) and Ω2(g) characterize the standard sphere.
We prove a lower bound for the first eigenvalue of the sub-Laplacian on sub-Riemannian manifolds with transverse symmetries. When the manifold is of H-type, we obtain a corresponding rigidity result: If the optimal lower bound for the first eigenvalue is reached, then the manifold is equivalent to a 1 or a 3-Sasakian s…
New estimators outperform maximum likelihood without hyper-parameter estimation.
problem Improving system identification performance without hyper-parameter estimation.
method Developed generalized Bayes and closed-form biased estimators using excess MSE.
result New estimators have comparable performance to empirical-Bayes-based regularized estimator.
Dual Bayesian Affine Estimators for Wiener-type state-space models
problem Estimating parameters in Wiener-type state-space models
method Fixed-point architecture combining two affine estimators
result Dual basis-parameter estimator achieves comparable parameter MSE to purely affine estimator
New estimator reduces kernel mean estimation error.
problem Kernel mean estimation in reproducing kernel Hilbert spaces.
method Corrupt data with known distributions and estimate kernel mean under the corrupted distribution.
result The marginalized kernel mean estimator achieves lower estimation error.
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 robust estimators for GANs under Wasserstein contamination.
problem Robust estimation of distributions under contamination.
method Wasserstein GAN-based estimators for location, covariance, and regression.
result Proposed estimators are minimax optimal in many scenarios.
New framework converts offline to online estimation using black-box offline estimators.
problem Convert offline estimation algorithms to online estimation algorithms.
method Oracle-Efficient Online Estimation (OEOE) framework.
result Achieves near-optimal online estimation error via black-box offline estimators.
Proposes variational autoencoder for efficient MMSE estimation.
problem Efficient parameterized MMSE estimation for noisy observations.
method Variational autoencoder models data distribution, approximates MMSE.
result Proposed estimator performs well compared to state-of-the-art.
Paper improves Fisher information estimation methods.
problem Estimating Fisher information for location parameters.
method Revisits and improves Bhattacharya estimator, introduces clipped estimator.
result Clipped estimator shows superior convergence rates in Gaussian noise.
Proposes a robust estimator for RD designs.
problem Estimating treatment effects in RD designs.
method Doubly robust estimator combining two estimators.
result Enhances robustness of treatment effect estimators.
New estimator reduces variance in discrete random variables.
problem Estimating gradients for discrete random variables with reduced variance.
method Sampling without replacement and Rao-Blackwellization.
result Our estimator is the most consistent gradient estimator across different entropy settings.
New risk-averse estimators uniquely characterize MAP and Wallace-Freeman estimators.
problem Formalizing and characterizing Bayesian point estimators.
method Formulated axioms for inference, showing unique characterizations of MAP and Wallace-Freeman estimators.
result Axioms uniquely characterize MAP and Wallace-Freeman estimators for different types of estimation problems.
SCOPE estimator improves covariance and precision matrix estimation.
problem Estimating covariance and precision matrices accurately.
method Distributionally robust optimization with convex spectral divergence.
result SCOPE estimator reduces spectral bias and improves condition number.
We present a multi-task learning approach to jointly estimate the means of multiple independent data sets. The proposed multi-task averaging (MTA) algorithm results in a convex combination of the single-task maximum likelihood estimates. We derive the optimal minimum risk estimator and the minimax estimator, and show t…
Obtaining more accurate equity value estimates is the starting point for stock selection, value-based indexing in a noisy market, and beating benchmark indices through tactical style rotation. Unfortunately, discounted cash flow, method of comparables, and fundamental analysis typically yield discrepant valuation estim…
We find an unbiased estimator for MMD variance.
problem Efficiently estimating the variance of MMD estimators.
method Extending and correcting previous work, we derive an unbiased estimator for MMD variance.
result We provide a truly unbiased estimator for MMD variance with no additional computational cost.
Stochastic volatility modelling of financial processes has become increasingly popular. The proposed models usually contain a stationary volatility process. We will motivate and review several nonparametric methods for estimation of the density of the volatility process. Both models based on discretely sampled continuo…
A new copula estimation method using classification.
problem Estimating copula density from joint and marginal distributions.
method Train a classifier to distinguish joint density from product of marginals.
result Empirically outperforms existing copula estimators.
This paper reviews SDR methods for multivariate response regression.
problem Handling sufficient dimension reduction for multivariate response regression.
method Characterizes SDR estimators as inverse or forward regression methods.
result Pooled marginal, projective resampling, distance-based, ordinary least squares, partial least squares, and semiparametric SDR estimators are discussed.
Density ratio estimation is a vital tool in both machine learning and statistical community. However, due to the unbounded nature of density ratio, the estimation procedure can be vulnerable to corrupted data points, which often pushes the estimated ratio toward infinity. In this paper, we present a robust estimator wh…
TAKDE optimizes kernel density estimation for real-time dynamic processes.
problem Real-time density estimation in applications like computer vision and signal processing.
method Derives asymptotic mean integrated squared error (AMISE) upper bound for 'sliding window' kernel density estimator and proposes TAKDE as a novel, theoretically optimal estimator.
result TAKDE outperforms other dynamic density estimators in terms of test log-likelihood and runtime.
Paper bridges score estimation to parameter and density estimation in DDPMs.
problem Efficiently estimating scores for generative models.
method Introduces a framework linking score estimation to parameter and density estimation.
result Denoising score-matching in DDPMs is asymptotically efficient for parameter estimation.
New method estimates density-derivative-ratios directly for clustering and ridge estimation.
problem Accurately estimating ratios of density derivatives.
method Direct estimation of density-derivative-ratios without density estimation.
result Developed methods significantly outperform existing techniques, especially for high-dimensional data.
We introduce two new estimators of the bivariate Hurst exponent in the power-law cross-correlations setting -- the cross-periodogram and local X-Whittle estimators -- as generalizations of their univariate counterparts. As the spectrum-based estimators are dependent on a part of the spectrum taken into consideration …
New estimator improves reliability of KL divergence estimation.
problem Estimating KL divergence reliably and efficiently.
method Proposes a new estimator using Reproducing Kernel Hilbert Space.
result Proposed estimator is consistent and more reliable for small datasets.
New method for fast volatility estimation robust to change points.
problem Robust high-frequency volatility estimation with change points.
method ℓ1-regularized power variation estimators using LARS for sparse estimation and dynamic programming for change point refinement.
result Minimax rates achieved for volatility estimators, providing accurate and smooth forecasts.
ROME improves density estimation for multi-modal, non-normal data.
problem Robust multi-modal density estimation in non-normal, highly correlated distributions.
method ROME uses clustering to segment multi-modal data into uni-modal clusters, then combines KDE estimates for each cluster.
result ROME outperforms state-of-the-art methods and is more robust to various distributions.
Paper introduces VDE, a variance-reduced determinant estimator.
problem Estimating determinants with low variance and efficiency.
method Combines variational inference and spherical normalizing flows.
result VDE achieves zero variance in ideal cases, requiring only one sample.
Private estimation of many quantiles using differential privacy.
problem Estimating quantiles of a distribution privately.
method Two approaches: 1) Private estimation of empirical quantiles, 2) Uniform density estimation.
result There is a tradeoff between estimating quantiles at specific points and uniformly estimating the quantile function.
New estimator improves mutual information estimation.
problem Estimating mutual information in data science and machine learning.
method Proposes a new estimator that uses a preliminary estimate of the data distribution.
result A preliminary estimate helps in estimating mutual information more accurately.
Kernel estimator improves spectral risk measure estimation.
problem Estimating spectral risk measures accurately.
method Kernel-based estimation of L-statistics for SRMs.
result Kernel estimator is strongly consistent and asymptotically normal.
Paper proposes robust LAD estimators for 2D sinusoidal model, proving consistency and normality.
problem Estimation of parameters in 2D sinusoidal models with outliers or heavy-tailed noise.
method Least absolute deviation (LAD) estimators for robust parameter estimation.
result Strong consistency and asymptotic normality of LAD estimators for 2D sinusoidal model parameters.
Combines multiple OPE estimators into a more accurate and efficient estimate.
problem Offline evaluation of recommender systems using biased data.
method Meta-analysis of correlated OPE estimators, accounting for inter-estimator correlation.
result Improved statistical efficiency and accuracy in estimating policy value.
New empirical Bayes estimator outperforms soft-thresholding for high-dimensional sparse vectors.
problem Estimating high-dimensional sparse vectors from noisy observations.
method Empirical Bayes shrinkage estimator using a Bernoulli-Gaussian prior.
result Hybrid estimator outperforms soft-thresholding in compressed sensing applications.
Unified framework for efficient estimation of unnormalized models.
problem Estimation of unnormalized models with statistical efficiency.
method Unified estimation framework combining density-ratio matching and nonparametric estimators.
result Asymptotic variance of proposed estimators is the same as MLE.
Paper introduces estimator response curve to assess mutual information estimators.
problem Assessing the performance of mutual information estimators.
method Utilizes estimator response curve to test various measures of association.
result Suboptimal estimators perform worse than optimal ones in real-world data.
Optimal and safe semi-supervised learning estimator for high-dimensional data.
problem Improving regression parameter estimation with unlabeled data in high-dimensional settings.
method Established minimax lower bound, proposed optimal and safe semi-supervised estimators.
result Optimal semi-supervised estimator achieves the minimax lower bound.