Improved bounds for estimating discrete distributions in KL divergence.
problem Estimating discrete distributions in KL divergence with accuracy.
method Used Laplace estimator and established concentration bounds.
result Deviation from mean scales as k / n \sqrt{k}/n k / n for n ≥ k n \ge k n ≥ k . New measure of robustness for estimators, with tight bounds for Gaussian mean estimation.
problem Developing robust statistical estimators for datasets with noise or outliers.
method Introducing empirical sensitivity as a new robustness measure and proving lower bounds for Gaussian mean estimation.
result Empirical sensitivity bounds for optimal estimators are tight, showing obstructions on mean and variance.
Jorge-Koutrofiotis and Pigola-Rigoli-Setti proved sharp sectional curvature estimates for extrinsically bounded submanifolds. Alias, Bessa and Montenegro showed that these estimates hold on properly immersed cylindrically bounded submanifolds. On the other hand, Alias, Bessa and Dajczer proved sharp mean curvature esti…
New lower bounds for private covariance estimation of Gaussian distributions are proven.
problem Proving tight lower bounds for private estimation tasks under differential privacy.
method Generalized fingerprinting method for exponential families and private Assouad method.
result Tight lower bounds for private covariance estimation in Frobenius and spectral norms.
Study non-asymptotic estimation bounds for LTI models with Gaussian noise.
problem Estimating parameters of LTI models with non-asymptotic error bounds.
method Sharp non-asymptotic lower bounds using Cramér-Rao and van Trees inequalities, concentration results, and differential geometric constructions.
result Sharp and rate-optimal lower bounds for mean square estimation risk.
Improves decision making by estimating bounds on potential outcomes.
problem Estimating individual treatment effects is complex and hard to estimate.
method Developed an algorithm to learn upper and lower bounds on potential outcomes that optimize an objective function defined by the decision maker.
result Our algorithm outperforms baselines, providing tighter, more reliable bounds.
Paper bounds subspace estimator error from noisy projections.
problem Estimating subspaces from noisy data.
method Derives perturbation bound on optimal subspace estimator.
result Fundamental result with implications in matrix completion and clustering.
Estimates for Schrödinger operators on manifolds with bounded Ricci curvature.
problem Quantifying unique continuation for Schrödinger operators on manifolds with specific curvature conditions.
method Proving quantitative unique continuation estimates for Schrödinger operators on manifolds with Ricci curvature bounded below.
result Upper bound for energy range and constant in terms of Ricci curvature and parameters of relatively dense set.
Improved bounds for discrete probability distribution estimation under the ℓ∞ norm.
problem Estimating discrete probability distributions under the ℓ∞ norm with improved bounds.
method Minimax bounds in expectation and high-probability tail bounds.
result Resolved open questions posed in Kontorovich and Painsky (JMLR, 2025), including a fully empirical tightest risk bound and identifying the worst-case extremal distribution.
Estimates mean curvature flow with geometric bounds.
problem Controlling mean curvature flow dynamics.
method Pointwise estimate using initial geometry and jHAj bound.
result Extension theorem and blowup rate estimate of HA.
The paper analyzes risk estimation methods and derives bounds for OCE risk.
problem Estimating the Optimized Certainty Equivalent (OCE) risk from samples.
method Derives mean-squared error and concentration bounds for SAA of OCE, and analyzes an efficient stochastic approximation-based estimator.
result Finite sample bounds and mis-identification probability bounds for the efficient estimator.
Lower bounds on private estimation of Gaussian covariance matrices.
problem Private estimation of Gaussian covariance matrices under various parameter regimes.
method Stein-Haff identity and fingerprinting lemma extensions.
result Lower bounds match existing upper bounds in the widest known parameters.
The paper provides gradient estimates for solutions on manifolds with integral Ricci bounds.
problem Global regularity estimates for solutions of Δ u = f Δu = f Δ u = f on Riemannian manifolds. method Proves L p L^p L p -gradient estimates under integral Ricci bounds and constructs a counterexample. result Optimal constant lower bounds on Ricci curvature are shown in the pointwise sense.
We estimate risk measures in Markov cost processes with lower and upper bounds.
problem Estimating risk measures in infinite-horizon discounted costs within Markov processes.
method Truncation scheme and lower/upper bounds for CVaR and variance estimation.
result Upper and lower bounds for CVaR and variance estimation match up to logarithmic factors.
The study analyzes weighted manifolds with curvature bounds, proving eigenvalue estimates and inequalities.
problem Analyzing geometric properties of weighted manifolds under Ricci curvature bounds.
method Develops geometric analysis techniques on weighted Riemannian manifolds with lower 0 0 0 -weighted Ricci curvature bounds. result Proves eigenvalue estimates for Steklov and ABP inequalities on weighted manifolds.
Recent research has made significant progress on the problem of bounding log partition functions for exponential family graphical models. Such bounds have associated dual parameters that are often used as heuristic estimates of the marginal probabilities required in inference and learning. However these variational est…
New MI bounds improve estimation in deep generative models.
problem Estimating mutual information without density information is intractable.
method Importance sampling, Annealed Importance Sampling, Generalized IWAE, MINE-AIS.
result Improved bounds for estimating mutual information in deep models.
This work introduces a new data-driven estimator for the Bayesian Cramér-Rao bound using score matching.
problem Benchmarking the performance of statistical estimators and providing a principled metric for system design and optimization.
method Score matching technique to estimate the Bayesian Cramér-Rao bound from training data.
result Developed novel non-asymptotic bounds on the score matching error and the Bayesian CRB estimator.
New robust estimators achieve subgaussian bounds using VC-dimension.
problem Robust estimation of sparse and corrupted data.
method Use of VC-dimension to measure statistical complexity.
result First robust estimators for sparse estimation with subgaussian rate.
Study compares nonsmooth spaces with integrable Ricci bounds.
problem Comparing geometric and functional inequalities on nonsmooth spaces.
method Localization method and one-dimensional comparison estimates.
result Extension of comparison principles to nonsmooth settings.
Estimates Bergman kernels on Kähler manifolds with Ricci bounds.
problem Estimate Bergman kernels on Kähler manifolds with given conditions.
method Using Liu-Székelyhidi's recent work and Kähler-Ricci flow.
result Weak asymptotic estimate for Bergman kernels.
The paper develops estimators for variance in graph structures using fused lasso.
problem Variance estimation in graph-structured problems.
method Developed linear time estimator for homoscedastic case and total variation regularization estimator for heteroscedastic case.
result Minimax rates and consistency for variance estimation in various graph structures.
Estimates Kähler metric diameters with entropy bound alone.
problem Estimating Kähler metric diameters.
method PDE techniques for L ∞ L^\infty L ∞ estimates of the Monge-Ampère equation, improving degeneracies. result Diameter bounds for Kähler-Ricci flow and Calabi-Yau manifolds.
Unified learning bound for covariate and concept shifts.
problem Generalization under distribution shift in machine learning.
method Support-agnostic definitions of covariate and concept shifts using entropic optimal transport, leading to a unified error bound applicable to various loss functions and label spaces.
result Development of estimators for shifts with concentration guarantees and the DataShifts algorithm for quantifying and estimating the error bound.
KSG mutual information estimator, which is based on the distances of each sample to its k-th nearest neighbor, is widely used to estimate mutual information between two continuous random variables. Existing work has analyzed the convergence rate of this estimator for random variables whose densities are bounded away fr…
Novel stability bounds for OT maps improve density estimation.
problem Estimating optimal transport maps between probability distributions.
method Developed novel stability bounds for OT maps, reducing the problem to density estimation.
result Stability bounds allow for sharper guarantees without smoothness assumptions.
Study on statistical estimation over Gaussian MAC, comparing analog and digital schemes.
problem Distributed minimax statistical estimation over a Gaussian MAC.
method Developed analog joint estimation-communication schemes and derived information-theoretic lower bounds.
result Achieved risk within a logarithmic factor of information-theoretic lower bounds.
New research sets the minimax lower bound for KSD estimation at sqrt(n).
problem Estimating goodness-of-fit using Kernel Stein Discrepancy (KSD) on high-dimensional spaces.
method Two complementary results proving the minimax lower bound of KSD estimation.
result The minimax lower bound of KSD estimation is n^(-1/2), indicating exponential difficulty with dimensionality.
Bidirectional bounds stabilize training of energy-based models.
problem Training energy-based models is difficult and prone to instability.
method Propose bidirectional bounds linking to gradient penalty and Jacobi-determinant estimator.
result Significant stabilization and high-quality density estimation achieved.
Sharp bounds on ATE with unmeasured confounders, valid even when misspecified.
problem Bounding average treatment effects with unmeasured confounders.
method Distributionally robust optimization, double sharpness, double validity.
result Proposes estimators with robustness properties for valid bounds.
Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.
problem Geometric and topological properties of Finsler metric measure manifolds with integral weighted Ricci curvature bounds.
method Establish Laplacian comparison theorem, volume comparison theorems, volume growth estimate, Gromov pre-compactness, local Dirichlet isoperimetric constant estimate.
result First Dirichlet eigenvalue estimate and gradient estimate for harmonic functions.
Study provides bounds for estimating intrinsic dimension using Gaussian kernels.
problem Estimating intrinsic dimension from data.
method Finite-sample concentration and anti-concentration bounds for Gaussian kernel sums.
result Explicit dependence on sample size, bandwidth, and geometric parameters.
The MDL two-part coding index of resolvability \textit{index of resolvability} index of resolvability provides a finite-sample upper bound on the statistical risk of penalized likelihood estimators over countable models. However, the bound does not apply to unpenalized maximum likelihood estimation or procedures with exceedingly small penalties. In this paper,…
Example surfaces with curvature bounds but no Laplacian eigenvalue lower bound.
problem Bounding curvature does not guarantee positive Laplacian eigenvalues.
method Explicit construction of surfaces with bounded curvature and diameter.
result Found surfaces without positive Laplacian eigenvalue lower bound.
The study establishes risk bounds for distributional regression estimators.
problem Estimating distributional regression models with nonparametric methods.
method Theoretical bounds for CRPS and MSE are derived for convex and non-convex constraints.
result Theoretical risk bounds are validated through experiments on simulated and real data.
Improved estimator for least squares using random projections achieves smaller error.
problem Improving the accuracy of least squares solutions for large-scale problems.
method James-Stein estimator applied to Gaussian sketching of least squares problems.
result Upper and lower bounds match when SNR is small and data matrix is well-conditioned.
Paper extends Steklov eigenvalue estimate to weighted graphs.
problem Steklov eigenvalue estimation on weighted graphs.
method Extended Perrin's estimate to general weighted graphs.
result Characterized rigidity of the extended estimate.
Lower bounds for eigenvalues on Bakry-Emery manifolds proven.
problem Eigenvalue estimates on Bakry-Emery manifolds.
method Generalised maximum principle and heat kernel estimates.
result Lower bounds for all eigenvalues proven.
New risk bound for drift estimator in stochastic models.
problem Theoretical guarantees for drift estimation in stochastic differential equations.
method Derives an explicit risk bound using diffusion model theory.
result Explicit decomposition of risk into multiple sources of error.
Sharp spectral gap estimates on manifolds with integral curvature bounds.
problem Proving spectral gap estimates on manifolds with integral curvature bounds.
method Generalizing previous results to include integral curvature bounds.
result Confirms a conjecture about spectral gap estimates on manifolds with integral curvature bounds.
We establish the first nonasymptotic error bounds for Kaplan-Meier-based nearest neighbor and kernel survival probability estimators where feature vectors reside in metric spaces. Our bounds imply rates of strong consistency for these nonparametric estimators and, up to a log factor, match an existing lower bound for c…
Eigenvalue estimates for Beltrami-Laplacian under specific curvature conditions.
problem Estimating eigenvalues of the Beltrami-Laplacian on manifolds with Bakry-Émery Ricci curvature.
method Using Bakry-Émery Ricci curvature conditions, the paper derives lower bounds for eigenvalues of the Beltrami-Laplacian.
result Lower bounds for eigenvalues of the Beltrami-Laplacian depend on curvature, gradient bounds, dimension, and diameter of the manifold.
Paper extends Bayesian Cramér-Rao bound with geometric considerations.
problem Estimation of covariance matrices with geometric structures.
method Intrinsic Bayesian Cramér-Rao bound with Riemannian geometry.
result Performance bounds for covariance matrix estimation.
The paper improves curvature estimates for Ricci flow solutions with bounded scalar curvature.
problem Proving curvature estimates for Ricci flow solutions with bounded scalar curvature.
method Localised weighted curvature integral estimates for solutions to Ricci flow.
result Integral curvature estimates imply a uniform bound on the spatial L 2 L^2 L 2 norm of the Riemannian curvature tensor. Estimates the first eigenvalue for embedded hypersurfaces in manifolds with Ricci curvature bounds.
problem Estimating the first eigenvalue for embedded hypersurfaces in manifolds with Ricci curvature bounds.
method Establishes a lower bound for the first nonzero eigenvalue of the Laplacian on embedded hypersurfaces in a closed oriented Riemannian manifold under Ricci curvature and sectional curvature bounds.
result The estimate depends on ambient curvature bounds, normal injectivity radius, and geometry of the hypersurface.
Estimates lower bounds for isoperimetric profiles and improves on previous estimates for specific manifolds.
problem Estimating lower bounds for isoperimetric profiles of specific Riemannian manifolds.
method Explicit lower bounds for isoperimetric profiles of Riemannian product manifolds.
result Improved lower bounds for isoperimetric profiles and Yamabe constants.
Vapnik-Chervonenkis (VC) dimension is a fundamental measure of the generalization capacity of learning algorithms. However, apart from a few special cases, it is hard or impossible to calculate analytically. Vapnik et al. [10] proposed a technique for estimating the VC dimension empirically. While their approach behave…
We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the L 2 L^2 L 2 Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions.