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.
Paper examines risk measure expansions under FGM dependence, improving accuracy at extreme levels.
problem Capturing higher-order tail behavior and dependence effects in risk measures.
method Second-order asymptotic expansions using extreme value theory and regular variation theory.
result Second-order approximations reduce approximation errors, especially at extreme confidence levels.
We develop and implement a novel fast bootstrap for dependent data. Our scheme is based on the i.i.d. resampling of the smoothed moment indicators. We characterize the class of parametric and semi-parametric estimation problems for which the method is valid. We show the asymptotic refinements of the proposed procedure,…
We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe…
Study characterizes conformal boundaries of de Sitter spacetimes.
problem Characterize conformal infinity of asymptotically de Sitter spacetimes.
method Derive constraints relating stress-energy tensor to conformal geometric data using higher conformal fundamental forms.
result Constraints on stress-energy tensor relate to conformal geometric data.
Study on Yang-Mills equations on conformally compact manifolds, finding obstructions and asymptotics.
problem Obstructing higher conformal Yang-Mills equations on conformally compact manifolds.
method Formal asymptotics, Dirichlet-to-Neumann maps, higher transverse derivative boundary operators.
result Obstructing current is the variation of a conformally invariant coefficient in the interior Yang-Mills energy expansion.
We present an extension of the Kolmogorov-Smirnov (KS) two-sample test, which can be more sensitive to differences in the tails. Our test statistic is an integral probability metric (IPM) defined over a higher-order total variation ball, recovering the original KS test as its simplest case. We give an exact representer…
The CGMY model's ATM call-price asymptotics are derived using characteristic function.
problem Deriving short-time asymptotics for the CGMY model's ATM call prices.
method Using the characteristic function, derived short-time asymptotics for the CGMY model's ATM call prices. Extracted higher-order coefficients by dynamic cutoff partitioning.
result Higher-order coefficients are derived for the CGMY model's ATM call prices.
The parametric complexity is the key quantity in the minimum description length (MDL) approach to statistical model selection. Rissanen and others have shown that the parametric complexity of a statistical model approaches a simple function of the Fisher information volume of the model as the sample size n goes to in…
Study describes how conformal metrics behave as Q-curvature changes, forming spherical bubbles.
problem Understanding the asymptotic behavior of conformal metrics with null Q-curvature.
method Analyzes the GJMS operator and uses higher order Bol's inequality to describe the metrics' behavior.
result Normalized conformal metrics form exactly one spherical bubble as λ approaches zero.
The paper proves existence and classification of translating solitons in warped product manifolds.
problem Existence and classification of translating solitons in warped product manifolds.
method Proving existence and classification results for translating solitons defined as initial conditions for higher order mean curvature flows in warped product manifolds.
result Existence and classification of translating solitons in warped product manifolds.
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.
The paper analyzes learning curves for kernel ridge regression with dot-product kernels.
problem Understanding the learning curves for different scaling regimes of data and model.
method Precise formulas for mean test error, bias, and variance in the mo∞ with m/dr constant regime. result A peak in the learning curve at m≈dr/r! for any integer r. 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.
We define a generalized mass for asymptotically flat manifolds using some higher order symmetric function of the curvature tensor. This mass is non-negative when the manifold is locally conformally flat and the σk curvature vanishes at infinity. In addition, with the above assumptions, if the mass is zero, then, nea…
Cross validation (CV) and the bootstrap are ubiquitous model-agnostic tools for assessing the error or variability of machine learning and statistical estimators. However, these methods require repeatedly re-fitting the model with different weighted versions of the original dataset, which can be prohibitively time-cons…
Study heat content in sub-Riemannian structures, proving asymptotic series existence and coefficients.
problem Analyzing heat content in sub-Riemannian manifolds.
method Adapting Savo's technique to sub-Riemannian structures, computing coefficients up to order 5.
result Existence of full asymptotic series and explicit computation of coefficients up to order 5.
Higher-order geometry modifies Newtonian dynamics and predicts anomalies in spacecraft motion.
problem Observing and understanding higher-order effects in general relativity.
method Generalizing the Einstein-Hilbert action to include higher-order infinitesimals and studying field equations and cosmologies.
result Higher-order corrections predict anomalies like the Pioneer and flyby effects.
We find complete hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of (elliptic) curvature functions which includes the higher order mean curvatures and their curvature quotients.
New framework for higher-order singular-value derivatives of rectangular matrices.
problem Challenging to derive higher-order Fréchet derivatives of singular values in real rectangular matrices.
method Using Kato's analytic perturbation theory for self-adjoint operators and embedding rectangular matrices into block self-adjoint operators.
result Closed-form expressions for the n-th order spectral variations of singular values. Unified asymptotic treatment for VaR- and expectile-based systemic risk measures.
problem Analyzing systemic risk measures under extreme system-wide disasters.
method Classified systemic risk measures into VaR- and expectile-based families, introduced new ICE and SICE measures, and provided second-order asymptotic results.
result Second-order asymptotics provide more accurate tail approximations for systemic risk measures.
It has been observed by Maldacena that one can extract asymptotically anti-de Sitter Einstein 4-metrics from Bach-flat spacetimes by imposing simple principles and data choices. We cast this problem in a conformally compact Riemannian setting. Following an approach pioneered by Fefferman and Graham for the Einstein e…
ROOT-SGD solves convex optimization problems with optimal nonasymptotic and near-optimal asymptotic performance.
problem Solving strongly convex and smooth unconstrained optimization problems using stochastic first-order algorithms.
method ROOT-SGD: Recursive One-Over-T SGD, averaging past stochastic gradients.
result Achieves state-of-the-art performance in both nonasymptotic and asymptotic senses.
The article models illiquid stocks using quantum calculus with asymptotic methods.
problem Modeling illiquid financial markets.
method Application of quantum stochastic calculus and asymptotic methods.
result Power series solutions can approximate quantum stochastic processes for longer time frames.
We introduce a class of "weakly asymptotically hyperbolic" geometries whose sectional curvatures tend to −1 and are C0, but are not necessarily C1, conformally compact. We subsequently investigate the rate at which curvature invariants decay at infinity, identifying a conformally invariant tensor which serves a…
Higher-order motif structures and multi-vertex interactions are becoming increasingly important in studies that aim to improve our understanding of functionalities and evolution patterns of networks. To elucidate the role of higher-order structures in community detection problems over complex networks, we introduce the…
Paper characterizes equilibrium strategies for stochastic control with higher-order moments.
problem Stochastic control problems with higher-order moments.
method Novel characterization of time-consistent control problems, deriving equilibrium conditions via BSDEs.
result Derives sufficient and necessary conditions for an open-loop Nash equilibrium control (ONEC) in a novel way.
New method improves model risk prediction using cross-audit projection.
problem Over-optimism in K-fold CV for binary classification. method Cross-audit projection (CAP) procedure combining resampling and asymptotic bias correction.
result CAP estimator achieves second-order asymptotic unbiasedness.
A solution of a problem by V.I.Arnol'd about higher analog of the asymptotic Hopf invariant of divergence-free vector fields is presented. A higher invariant of magnetic fields, which is not expressed from the asymptotic linking numbers of magnetic lines is constructed and examples of an asymptotic invariants is constr…
We show that the coercivity of the modified Ding functional leads to the existence of a certain kind of balanced metrics and their convergence to the Kähler-Ricci soliton modulo automorphisms. In our results, we do not assume that the vanishing of the higher order modified Futaki invariants introduced by Berman-Nyström…
Given a regular bounded domain Ω⊂R2m, we describe the limiting behavior of sequences of solutions to the mean field equation of order 2m, m≥1, (−Δ)mu=ρ∫Ωe2mudxe2muinΩ, under the Dirichlet boundary condition and the bound 0<ρ≤C. We emphasize the connection wi…
New tensors help determine if metrics are related to Poincaré-Einstein ones.
problem Determining when metrics on conformally compact manifolds are related to Poincaré-Einstein metrics.
method Developed new tensors and used conformal tractor calculus to analyze metrics.
result The vanishing of these new tensors is a necessary and sufficient condition for a metric to be related to a Poincaré-Einstein metric.
New method identifies structural parameters without assuming uncorrelated errors.
problem Identifying structural parameters in simultaneous equation models.
method Exploits higher-order cumulant restrictions, not requiring uncorrelated errors.
result Simple diagonality condition on hth-order cumulants identifies structural parameter matrix. Existence of singular gradient Ricci solitons proved in higher dimensions.
problem Proving the existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions.
method Fixed point argument to prove the existence of infinitely many solutions for the given equation.
result Infinitely many solutions for the equation 2r2h(r)hrr(r)=(n−1)h(r)(h(r)−1)+rhr(r)(rhr(r)−λr−(n−1)) are found. New method identifies causal direction with latent confounders.
problem Identifying causal direction in presence of multiple latent variables.
method Use of joint higher-order cumulant matrix properties.
result Causal asymmetry can be seen from rank deficiency properties of cumulant matrices.
Higher-dimensional spacetimes have well-behaved boundaries.
problem Understanding boundaries of higher-dimensional spacetimes.
method Analyzing (n+1)-dimensional Myers-Perry metrics at spacelike infinity. result Optimal conformal completion at spacelike infinity for Cn−3,1 differentiability class. WeSpeR speeds up non-linear shrinkage for high-dimensional weighted covariance.
problem Computing non-linear shrinkage formulas for high-dimensional weighted sample covariance.
method Derive extit{WeSpeR} algorithm using asymptotic sample spectrum properties.
result Significantly speeds up non-linear shrinkage in dimensions higher than 1000.
Understanding the asymptotic behavior of wide networks is of considerable interest. In this work, we present a general method for analyzing this large width behavior. The method is an adaptation of Feynman diagrams, a standard tool for computing multivariate Gaussian integrals. We apply our method to study training dyn…
Analyzes branch points of area-minimizing currents with non-2 planar frequency.
problem Understanding the structure of area-minimizing currents near branch points.
method Intrinsic frequency function and geometric arguments avoiding center manifolds.
result Establishes higher order asymptotics and topological control near branch points.
The paper refines classical covariance asymptotics using geometric information geometry.
problem Deviation of finite-sample behavior from classical predictions in curved models.
method Develops a curvature-aware refinement by viewing parametric families as Riemannian manifolds with Fisher-Rao metric.
result Derives an \(n^{-2}\) correction to the leading \(n^{-1}I(θ)^{-1}\) covariance term for score-root estimators.
Study uses renormalized area to determine metric expansion from minimal surfaces.
problem Recovering the expansion of asymptotically hyperbolic metrics from minimal surfaces.
method Uses renormalized area functional on minimal submanifolds to recover metric expansion.
result Proves rigidity for log-analytic metrics and determines obstruction tensor.
We analyze analytic approximation formulae for pricing zero-coupon bonds in the case when the short-term interest rate is driven by a one-factor mean-reverting process with a volatility nonlinearly depending on the interest rate itself. We derive the order of accuracy of the analytical approximation due to Choi and Wir…
Sharp inequalities for radial functions on hyperbolic spaces without boundary conditions.
problem Establishing inequalities for radial functions on hyperbolic spaces without zero boundary conditions.
method Novel approach considering both bounded and unbounded domains, focusing on weighted Sobolev and Adams-Trudinger-Moser embeddings.
result Theorems 1.2, 1.3, and 1.4 for weighted Sobolev embedding theorems, and Theorems 1.5 and 1.6 for Adams-Trudinger-Moser type embedding theorems.
Counting meanders on surfaces of arbitrary genus, with precise asymptotics.
problem Counting and understanding meanders on surfaces of arbitrary genus.
method Square-tiled surfaces, moduli spaces of Abelian and quadratic differentials, Witten-Kontsevich 2-correlators.
result Asymptotic probability and polynomial growth of meanders with intersections.
Paper extends foliation results in higher dimensions for Schwarzschild spaces.
problem Existence of foliations by constant harmonic mean curvature hypersurfaces in asymptotically Schwarzschild manifolds.
method Generalization to higher dimensions, proving existence under arbitrary dimensionality.
result Existence of foliations by constant harmonic mean curvature hypersurfaces in asymptotically Schwarzschild manifolds of arbitrary dimension.
The paper calculates heat kernel and closed geodesic asymptotics for nilpotent coverings.
problem Heat kernel and closed geodesic asymptotics for nilpotent coverings.
method Finite-dimensional rational Floquet-Bloch theory, Pytlik functional, and spectral sums.
result Genuinely local, pointwise higher-order heat-kernel expansions.
We give a surgery formula for the asymptotic behavior of the sequence given by the logarithm of the higher dimensional Reidemeister torsion. Applying the resulting formula to Seifert fibered spaces, we show that the growth of the sequences has the same order as the indices and we give the explicit values for the limits…
We derive asymptotic expansions for option data to detect infinite variation volatility.
problem Detecting infinite variation volatility in high-frequency option data.
method Nonparametric higher-order asymptotic expansions for small-time changes of characteristic functions of Itô semimartingales.
result Evidence of infinite variation volatility in high-frequency option data.