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.
Estimation of density derivatives is a versatile tool in statistical data analysis. A naive approach is to first estimate the density and then compute its derivative. However, such a two-step approach does not work well because a good density estimator does not necessarily mean a good density-derivative estimator. In t…
New estimator for estimating derivatives in nonparametric regression.
problem Estimating derivatives of regression functions.
method Plug-in kernel ridge regression (KRR) estimator.
result Plug-in property for derivatives estimation, optimal rate of convergence.
Estimates cross-impact on derivatives markets using E-Mini futures and options.
problem Empirical estimation of cross-impact on complex financial instruments like derivatives.
method Modeling derivatives prices as a function of stochastic factors and trades on both factors and derivatives.
result Simple framework successfully captures cross-impact on derivatives markets.
Derives Hessian estimates for Lagrangian mean curvature equation.
problem Lagrangian mean curvature equation with supercritical phase and bounded second derivatives.
method Derives a priori interior Hessian estimates.
result Hessian estimates for Lagrangian mean curvature equation.
Estimates smooth functions and their derivatives from noisy data.
problem Estimating smooth functions and their derivatives from noisy data.
method Least squares estimators and minimizers of smoothness subject to error bounds.
result Consistent estimators with convergence rates as n increases.
Derivative estimates for pluriclosed flow control curvature and torsion.
problem Deriving derivative estimates for the pluriclosed flow.
method Control higher order derivatives of Chern curvature and torsion using Chern curvature; derive an estimate for torsion tensor using Chern Ricci curvature in dimension two; find a monotonic quantity in Hermitian-symplectic case.
result All Hermitian-symplectic solitons are Kähler Ricci solitons.
Conditional Leibniz Derivative Estimation reduces variance in stochastic models.
problem Estimating derivatives in stochastic models with discontinuous sample performance.
method Combining push-out likelihood ratio method with Leibniz integral rules.
result Conditional Leibniz estimator reduces variance and is easy to implement.
Modes and ridges of the probability density function behind observed data are useful geometric features. Mode-seeking clustering assigns cluster labels by associating data samples with the nearest modes, and estimation of density ridges enables us to find lower-dimensional structures hidden in data. A key technical cha…
In this short note we present local derivative estimates for heat equations on Riemannian manifolds following the line of W.-X. Shi. As an application we generalize a second derivative estimate of R. Hamilton for heat equations on compact manifolds to noncompact case.
Study bounds derivatives of solutions to a specific equation on domains.
problem Bounding second derivatives of solutions to the σk-Yamabe equation. method Proves local pointwise second derivative estimates for positive W2,p solutions. result Establishes bounds for derivatives of solutions to the σk-Yamabe equation. The article derives a novel Gram-Charlier A (GCA) Series based Extended Rule-of-Thumb (ExROT) for bandwidth selection in Kernel Density Estimation (KDE). There are existing various bandwidth selection rules achieving minimization of the Asymptotic Mean Integrated Square Error (AMISE) between the estimated probability d…
The paper derives new gradient and Hessian estimates for nonlinear parabolic equations.
problem Estimating solutions to nonlinear weighted parabolic equations.
method Derives Li-Yau and Hamilton type gradient estimates, and Hessian estimates.
result New gradient and Hessian estimates for positive solutions of nonlinear parabolic equations.
Paper develops methods to estimate derivative of dose-response curve for continuous treatments.
problem Estimating the derivative of the dose-response curve for continuous treatments.
method Doubly robust (DR) inference method using kernel smoothing, bias-corrected IPW and DR estimators.
result Proposes novel bias-corrected IPW and DR estimators for continuous treatments.
A typical goal of supervised dimension reduction is to find a low-dimensional subspace of the input space such that the projected input variables preserve maximal information about the output variables. The dependence maximization approach solves the supervised dimension reduction problem through maximizing a statistic…
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…
Paper introduces efficient methods for estimating cross-partial derivatives and sensitivity indices.
problem Efficiently estimating cross-partial derivatives and sensitivity indices in complex models.
method Using randomized points and constraints, the paper develops estimators with optimal convergence rates and low bias.
result The estimators achieve optimal rates of convergence and do not suffer from the curse of dimensionality.
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.
Optimal estimator derived for partially observable LTI systems.
problem Optimal estimator for partially observable LTI systems.
method State-space representation for derivation of optimal estimator.
result Derivation of minimum error variance estimator for partially observable LTI systems.
We present a method to derive local estimates for some classes of fully nonlinear elliptic equations. The advantage of our method is that we derive Hessian estimates directly from C0 estimates. Also, the method is flexible and can be applied to a large class of equations.
We estimate Radon-Nikodym derivatives using regularization in reproducing kernel Hilbert spaces.
problem Estimating Radon-Nikodym derivatives in various applications.
method General regularization scheme in reproducing kernel Hilbert spaces.
result High order accuracy in reconstructing Radon-Nikodym derivatives at any point.
We give a maximum principle proof of interior derivative estimates for the Kähler-Ricci flow, assuming local uniform bounds on the metric.
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.
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.
The article derives gradient estimations for semilinear equations on geometric flows.
problem Gradient estimation for semilinear equations on geometric flows.
method Derives both Hamilton and Souplet-Zhang type gradient estimations.
result Gradient estimations for semilinear equations on geometric flows.
A new gradient estimator for categorical distributions reduces bias and variance.
problem Intractability of gradients for categorical distributions in discrete latent variable models.
method CatLog-Derivative trick and IndeCateR gradient estimator.
result IndeCateR reduces bias and variance of gradients for categorical distributions.
Backpropagation and the chain rule of derivatives have been prominent; however, the total derivative rule has not enjoyed the same amount of attention. In this work we show how the total derivative rule leads to an intuitive visual framework for creating gradient estimators on graphical models. In particular, previous …
In this paper, we derive apriori estimates for constant scalar curvature Kähler metrics on a compact Kähler manifold. We show that higher order derivatives can be estimated in terms of a C0 bound for the Kähler potential. We also discuss some local versions of these estimates which can be of independent interest.
Derives Li & Yau estimates for heat equations on manifolds.
problem Analyzing positive solutions of semilinear heat equations on manifolds.
method Adapts Li & Yau estimates to derive new inequalities.
result Derives Harnack inequality and discusses monotonicity, convexity, decay estimates.
New method uses DistRL to estimate entire payoff distribution for financial derivatives.
problem Traditional methods focus on expected option value; this tackles risk-aware pricing.
method Reinterprets and proposes a framework using Distributional Reinforcement Learning (DistRL).
result Demonstrates enhanced risk-aware pricing and uncertainty quantification on Asian options.
New rigidity estimate derived via harmonic map flow.
problem Rigidity of maps from S2 to S2. method Harmonic map flow approach.
result Rigidity estimate derived.
This paper precisely estimates transformer derivatives for explicit learning guarantees.
problem Computing fully-explicit generalization bounds for transformers with precise higher-order derivative estimates.
method Analyzes and estimates all higher-order derivatives of transformers with multiple attention heads and layer normalization.
result Obtains explicit pathwise generalization bounds for transformers learning from non-i.i.d. samples.
Introduces conformal Bach flow and proves its well-posedness and backward uniqueness.
problem Analyzing the long-time behavior of conformal Bach flow.
method Establishes well-posedness and backward uniqueness; derives L2-estimates of curvatures. result Derives Shi's pointwise-estimate of derivatives of curvatures without assuming Sobolev constant bound.
Derives L∞ estimate for Kähler-Ricci flows with weaker conditions.
problem Estimating solutions to Kähler-Ricci flows under weaker conditions.
method Extends recent techniques to more general geometric cases.
result Derives L∞ estimate for Kähler-Ricci flows with weaker conditions. In this note, we obtain the asymptotic estimate for the time derivative of the Φ-entropy in terms of the lower bound on the Bakry-Emery Γ2 curvature. In the cases of Hyperbolic space and Heisenberg group, we show that the time derivative of the Φ-entropy is non-increasing, and we also get sharp asymptotic bound …
Paper studies M-estimators with derivatives and residual distribution for robust adaptive tuning.
problem Tackles robustness and adaptive tuning of M-estimators with heavy-tailed noise.
method Provides formulae for derivatives, characterizes residual distribution, proposes adaptive criterion.
result Characterizes distribution of residuals and proposes adaptive criterion as out-of-sample error proxy.
Eigenvalue estimates for weighted manifolds with applications.
problem Eigenvalue estimates for weighted Riemannian manifolds.
method Derivation of various eigenvalue estimates for the Hodge Laplacian acting on differential forms.
result Derivation of an inequality relating eigenvalues of the Jacobi operator and the spectrum of the Hodge Laplacian.
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. Optimizes heat equation estimates on noncompact manifolds.
problem Improving gradient estimates for heat equations on noncompact manifolds.
method Localized and global noncompact versions of Hamilton's gradient estimate for positive solutions to the heat equation.
result Essentially optimal estimates significantly improve previous results.
The paper derives subgradient estimates for a specific nonlinear subparabolic equation on pseudo-Hermitian manifolds.
problem Deriving subgradient estimates for positive solutions to a nonlinear subparabolic equation on pseudo-Hermitian manifolds.
method Using the CR sub-Laplacian comparison property, the paper derives local subgradient estimates for positive solutions to the given equation.
result The paper establishes subgradient estimates for positive solutions to the nonlinear subparabolic equation.
Derives gradient estimates for CR heat equation on pseudo-Hermitian manifolds.
problem Estimating solutions to CR heat equation on complex manifolds.
method Local and global Li-Yau type gradient estimates.
result Gradient estimates and Harnack inequality for positive solutions.
Derives concavity inequality and estimates for k-Hessian equations.
problem Interior estimates and curvature estimates for k-Hessian equations. method Concavity inequality and semi-convexity condition.
result Interior estimates and Liouville-type result for semi-convex solutions.
Important information concerning a multivariate data set, such as clusters and modal regions, is contained in the derivatives of the probability density function. Despite this importance, nonparametric estimation of higher order derivatives of the density functions have received only relatively scant attention. Kernel …
Derives gradient estimation for a specific heat equation on evolving manifolds.
problem Gradient estimation for a generalized heat equation on evolving weighted Riemannian manifolds.
method Derives gradient estimation for a specific heat equation on evolving weighted Riemannian manifolds.
result Derives a Harnack type inequality and a Liouville type theorem as applications of gradient estimation.
Developed moment estimators for affine stochastic volatility models.
problem Estimating parameters of affine stochastic volatility models.
method Introduced recursive equations for moments and proposed moment estimators.
result Established a central limit theorem and derived asymptotic covariance matrix.
We derive an unbiased estimator for expectations over discrete random variables based on sampling without replacement, which reduces variance as it avoids duplicate samples. We show that our estimator can be derived as the Rao-Blackwellization of three different estimators. Combining our estimator with REINFORCE, we ob…
Optimizes shortfall risk using gradient-based methods.
problem Optimizing utility-based shortfall risk measures.
method Gradient-based stochastic optimization, non-asymptotic bounds derivation.
result Non-asymptotic convergence rate for optimizing UBSR.
Quantum advantage in derivative pricing requires 8k qubits and 54M T-depth.
problem Quantum advantage in pricing derivatives.
method Re-parameterization method combining pre-trained variational circuits and fault-tolerant quantum computing.
result Benchmark use cases require 8k logical qubits and a T-depth of 54 million.