Abstract: Geometrically reformulates estimation theory for finite-dimensional C*-algebras.
problem Estimation theory for finite-dimensional C*-algebras.
method Geometrical formulation of estimation theory.
result Derivation of Cramer-Rao and Helstrom bounds.
The need to estimate smooth probability distributions (a.k.a. probability densities) from finite sampled data is ubiquitous in science. Many approaches to this problem have been described, but none is yet regarded as providing a definitive solution. Maximum entropy estimation and Bayesian field theory are two such appr…
New theory of sensitivity for unbiased estimators using Wasserstein geometry.
problem Estimating the instability of estimators under small perturbations.
method Developed a new theory based on Wasserstein geometry, analogous to classical Cramér-Rao theory.
result Wasserstein-Cramér-Rao lower bound for sensitivity of unbiased estimators.
A framework for analyzing regularizers to ensure trustworthy theory-driven model estimation.
problem Uncertain choice of regularizers can compromise the interpretability of deep grey-box models.
method Adapting neural net architecture and training objective to analyze regularizer behavior empirically.
result Empirical analysis of regularizers helps in making a justified choice for trustworthy theory-driven model estimation.
New proof for stability estimates in complex equations without pluripotential theory.
problem Stability estimates for complex Monge-Ampère and Hessian equations.
method New proof using general degenerations of background metrics.
result Uniform stability estimates for both equations under various degenerations.
Develops asymptotic theory for adversarial estimators.
problem Estimating unknown functions in machine learning and econometrics.
method Derives convergence rates and normality of A-estimators under various conditions.
result Normality of neural-net M-estimators, overcoming previous technical issues.
The paper applies potential theory to conformal geometry, proving theorems and dimension estimates.
problem Understanding the behavior of solutions near singularities in conformal geometry.
method Linear and nonlinear potential theory applied to conformal geometry problems.
result Established Huber's type theorems and Hausdorff dimension estimates for conformal geometry.
This work broadens optimal transport map estimation theory to stochastic settings.
problem Existing theory for optimal transport map estimation is restricted to deterministic maps under specific conditions.
method Introduces a novel metric for evaluating stochastic maps, develops computationally efficient estimators with robust guarantees.
result First general-purpose theory for map estimation compatible with real-world stochastic applications.
Study connects covariance cleaning theory to information theory for heavy-tailed distributions.
problem Optimizing covariance matrices for heavy-tailed distributions using information theory.
method Minimizing Frobenius norm and information loss between true and estimated covariance matrices.
result Asymptotic regime of large matrices minimizes information loss for Student's t distributions.
Extends width estimates to family case using index theory.
problem Sharp width estimates for Riemannian bands with positive scalar curvature.
method Dirac operators and family index theory.
result Proves width estimate for fiber bundles with infinite A-hat area.
Develops a statistical framework for coherent risk estimation.
problem Constructing coherent risk estimators with sound financial and statistical properties.
method Inspired by axiomatic risk measure theory, defines coherent risk estimators through robust representations linked to L-estimators. result Demonstrates that coherence of a risk measure does not necessarily carry over to its estimators and shows alternative weight structures can lead to different outcomes.
In this paper, we develop a general theory of truncated inverse binomial sampling. In this theory, the fixed-size sampling and inverse binomial sampling are accommodated as special cases. In particular, the classical Chernoff-Hoeffding bound is an immediate consequence of the theory. Moreover, we propose a rigorous and…
Paper proves rigidity estimates for hyperbolic shells and applies them to \(Γ\)-limit theory.
problem Rigidity of hyperbolic shells and their \(Γ\)-limit behavior.
method Nonlinear rigidity estimates for \(H^1\) deformations and hyperbolic shells with clamped lateral boundary.
result Derives the optimal exponent \(h^{-4/3}\) for hyperbolic shells.
New method uses extreme value theory to estimate neural network errors.
problem Quantifying the error of neural networks, especially for large values.
method Applying extreme value theory to approximate the distribution of error.
result Developed a new estimator for the shape parameter of the Pareto distribution.
Estimates the rational homological dimension of Riemann surfaces with boundary and marked points.
problem Estimating the homological dimensions of Riemann surfaces with boundary and marked points.
method Developed an estimate for the rational homological dimension of Riemann surfaces with possible boundary and marked points.
result Provided an estimate for the rational homological dimension of Riemann surfaces with boundary and marked points.
Unified theory for causal inference using various methods.
problem Estimating causal effects in ATE estimation.
method Riesz regression, covariate balancing, DRE, TMLE, matching estimator.
result Unified theory integrating multiple methods for ATE estimation.
We give an overview of the generalized Calderón-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted estimates for such operators and their commutators with $\BMO$ functions. Lp−Lq of…
We present a theory of homogeneous volatility bridge estimators for log-price stochastic processes. The main tool of our theory is the parsimonious encoding of the information contained in the open, high and low prices of incomplete bridge, corresponding to given log-price stochastic process, and in its close value, fo…
In a wide variety of sequential decision making problems, it can be important to estimate the impact of rare events in order to minimize risk exposure. A popular risk measure is the conditional value-at-risk (CVaR), which is commonly estimated by averaging observations that occur beyond a quantile at a given confidence…
Sharp L∞ estimates proved for complex Monge-Ampère equations.
problem Proving sharp L∞ estimates for complex Monge-Ampère equations. method PDE proof covering fixed and degenerating background metrics, extends to general fully non-linear equations.
result Sharp L∞ estimates proved for complex Monge-Ampère equations. New bounds for non-convex estimators without Bernstein condition.
problem Sharp excess risk bounds for non-convex and improper estimators.
method Exponential-tail local Rademacher complexity risk bounds with offset condition.
result Sharp bounds for non-convex and improper estimators without Bernstein condition.
InfoBridge uses diffusion bridges to estimate mutual information accurately.
problem Estimating mutual information between random variables.
method Formulated mutual information estimation as a domain transfer problem using diffusion bridge models.
result Demonstrated unbiased estimator for various data types.
New DL algorithm estimates OFDM channels without pilots.
problem Estimating OFDM channels in deep fading conditions.
method Deep learning (DL) for blind channel estimation.
result First theory on MSE performance of DL-based estimator.
We introduce a covariance matrix estimator that both takes into account the heteroskedasticity of financial returns (by using an exponentially weighted moving average) and reduces the effective dimensionality of the estimation (and hence measurement noise) via techniques borrowed from random matrix theory. We calculate…
Consistent estimator derived for confounding strength in observational data.
problem Estimating confounding strength in observational data is challenging due to unobserved confounders.
method Derived and adapted a consistent estimator using tools from random matrix theory.
result The original estimator is not consistent, but an adapted one is.
Develop a comprehensive theory for regularized M-estimation in reproducing kernel Hilbert spaces.
problem Regularized M-estimation in reproducing kernel Hilbert spaces
method Existence and measurability of the estimator, sharp rates of convergence
result New rates for tensor product Sobolev spaces
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.
Paper introduces a new power-dominance axis in estimator design.
problem Estimator design trade-off between bias and variance.
method Introduces a third power regime, `power-dominant', with an unavoidable error penalty.
result Any estimator in the `power-dominant' regime is structurally sub-optimal.
A new distance metric derived from information theory and estimation theory.
problem Developing a robust distance metric for complex signal distributions.
method Information-Estimation Metric (IEM) derived from continuous probability density and denoising errors.
result The IEM is a valid global distance metric that adapts to the geometry of complex distributions.
Paper develops a new estimator for high-dimensional panel data with common shocks.
problem Cross-sectionally dependent errors driven by common shocks in high-dimensional panel data.
method Factor-augmented sparse-group LASSO estimator combining MIDAS aggregation with latent factors.
result The estimator outperforms standard LASSO for prediction and estimation in settings with cross-sectional dependence.
Develops pathwise analysis for log-optimal portfolios using rough paths theory.
problem Analyzing stability and approximation of log-optimal portfolios.
method Pathwise approach based on càdlàg rough paths theory.
result Establishes pathwise stability and error estimates for log-optimal portfolios.
The study provides statistical theory for WGANs in time series forecasting.
problem Statistical analysis of WGANs for time series forecasting.
method Statistical theory and upper bounds for excess Bayes risk, weak convergence, and confidence intervals.
result Developed confidence intervals for time series forecasting using WGANs.
Paper improves risk estimation for extreme events.
problem Estimating extreme risks accurately.
method Modified Bayes risk for expectiles, asymptotic expansions, efficient estimators.
result Asymptotic normality of estimators proved.
FQE with deep neural networks achieves asymptotic normality and finite-sample bounds.
problem Theoretical understanding of FQE with general differentiable function approximators.
method Z-estimation theory applied to FQE with deep neural networks.
result FQE estimation error is asymptotically normal with explicit variance.
The paper develops AMP theory for sparse and robust regression with polynomial iterations.
problem Challenges in high-dimensional statistical estimation due to asymptotic theory breakdown.
method Non-asymptotic distributional theory of AMP for sparse and robust regression.
result First finite-sample non-asymptotic distributional theory of AMP for polynomial iterations.
We present a unified framework for low-rank matrix estimation with nonconvex penalties. We first prove that the proposed estimator attains a faster statistical rate than the traditional low-rank matrix estimator with nuclear norm penalty. Moreover, we rigorously show that under a certain condition on the magnitude of t…
We present a comprehensive theory of homogeneous volatility (and variance) estimators of arbitrary stochastic processes that fully exploit the OHLC (open, high, low, close) prices. For this, we develop the theory of most efficient point-wise homogeneous OHLC volatility estimators, valid for any price processes. We intr…
This paper provides performance guarantees for neural estimation of statistical distances.
problem Developing performance guarantees for neural estimation of statistical distances.
method Non-asymptotic error bounds using function approximation theorems and empirical process theory.
result Established a fundamental tradeoff between approximation and estimation errors in neural estimation of statistical distances.
We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in R2 introduced by Rivière. Applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric PDE theory, including some of the theory of harmoni…
This work develops a learning theory for inferring interaction kernels in complex agent systems.
problem Modeling complex interactions in systems of particles or agents.
method Nonparametric regression and approximation theory.
result Strong consistency and optimal convergence rates for estimators of interaction kernels.
The paper examines skill estimation and variance under model misspecification in IRT.
problem Underestimation and overestimation of skills when non-compensatory model is misspecified as compensatory.
method Theoretical approach to analyze underestimation and overestimation of skills and variance.
result Overestimation of skills occurs around the origin and asymptotic variance differs under model misspecification.
Paper analyzes singular subspace estimation in noisy matrix models.
problem Estimating low-rank signals in noisy matrix data.
method Asymptotic distributional theory, extreme value theory, saddle point approximation, random matrix theory.
result Plug-in test statistic based on two-to-infinity norm has higher power for detecting structured alternatives.
The problem of f-divergence estimation is important in the fields of machine learning, information theory, and statistics. While several nonparametric divergence estimators exist, relatively few have known convergence properties. In particular, even for those estimators whose MSE convergence rates are known, the asympt…
Novel mutual information bound improves statistical inference rates.
problem Improving statistical inference rates in Bayesian nonparametrics.
method Introduces a novel mutual information bound.
result Improved contraction rates for fractional posteriors.
Gaussian and bootstrap methods improve ATE estimator accuracy.
problem Improving the accuracy of Average Treatment Effect (ATE) estimators.
method Gaussian approximation and bootstrap procedures.
result Precise bounds on ATE estimator accuracy quantifying key parameters.
Gem theory helps estimate trisection genus of 4-manifolds.
problem Estimating the trisection genus of 4-manifolds.
method Using gem theory, a type of edge-colored graphs dual to colored triangulations.
result Regular genus is an upper bound for trisection genus of closed 4-manifolds.
The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Koda…
New sampling method estimates Shapley values more accurately.
problem Exponential time complexity of computing Shapley values.
method Multilinear sampling algorithm based on game theory.
result Our method reduces variance and provides more accurate Shapley value estimations.