In this paper, we give a proof of the quantitative Morse theorem stated by {Y. Yomdin} in \cite{Y1}. The proof is based on the quantitative Sard theorem, the quantitative inverse function theorem and the quantitative Morse lemma.
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48 results for “quantitative inverse function theorem”
This paper studies neural network operators and their convergence properties.
problem Understanding the approximation and convergence of neural network operators.
method Proves density results, convergence estimates, and Voronovskaya-type theorems.
result Establishes quantitative convergence estimates and derives Voronovskaya-type theorems.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
Paper proves mass theorems for nonnegative scalar curvature metrics.
problem Proving mass theorems for metrics with nonnegative scalar curvature.
method New local inverse mean curvature flow with quantitative stability.
result Existence of isoperimetric sets in low regularity metrics.
The paper proves a generalized inverse function theorem for curved spaces.
problem Proving a generalized inverse function theorem for curved spaces.
method Obstruction theory for homomorphisms and homotopy transfer theorem for curved algebras.
result A morphism of curved spaces which is a quasi-isomorphism at a point has a local homotopy inverse.
The study proves a quantitative functional CLT for neural networks with smooth activation functions.
problem Understanding the convergence rates of neural networks with different activation functions.
method Functional versions of the Stein-Malliavin approach and a quantitative functional central limit theorem.
result Rates of convergence depend on the smoothness of the activation function, ranging from logarithmic to sqrt(n).
The paper proves bounds on curvature and injectivity radius for convex sums of Riemannian metrics.
problem Understanding the geometry of convex sums of Riemannian metrics.
method Quantitative inverse function theorem and Riemannian geometry techniques.
result Injectivity radii of convex sums have uniform lower bounds.
Proves theorem for Riemannian manifolds, extending previous work.
problem Proving Quantitative Fatou Theorem on Riemannian manifolds.
method Extending ε-approximation lemma to manifold setting.
result Proves Quantitative Fatou Theorem for Lipschitz domains on Riemannian manifolds.
Study shows stability of Schrödinger operator spectral data on a manifold.
problem Determining a manifold and potential function from spectral data.
method Approximation of spectral data on a subset to determine manifold and potential.
result Quantitative stability estimate for Schrödinger operator inverse problem.
Paper shows stability of metric reconstruction for orbifolds from spectral data.
problem Determining the metric structure of collapsing orbifolds from spectral data.
method Improved quantitative unique continuation for wave operator on Riemannian manifolds.
result Quantitative stability of inverse problem for Riemannian orbifolds.
Abstracts a theorem for non-smooth maps in infinite dimensions.
problem Generalizing inverse mapping theorem for non-smooth maps.
method Introduces property A and applies it to non-smooth maps.
result Generalized inverse mapping theorems for non-smooth maps.
Inverse function theorem and homotopy description for L-infinity bundles.
problem Inverse function theorem and homotopy description for L-infinity bundles.
method Local sections composed of elementary morphisms.
result Simple description of homotopy category of L-infinity bundles.
Quantitative Obata's Theoremmath.FA
We prove a quantitative version of Obata's Theorem involving the shape of functions with null mean value when compared with the cosine of distance functions from single points. The deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise. These results are obtained in the genera…
Quantum neural networks can approximate noisy functions accurately.
problem Approximating noisy functions with quantum neural networks.
method Universal approximation theorem with error bounds for noisy quantum neural networks.
result Quantum neural networks can approximate noisy functions with precise error bounds.
The Stone-Weierstrass theorem aids in solving inverse problems on specific manifolds.
problem Inverse problems on holomorphically separable Kähler manifolds and conformally transversally anisotropic manifolds.
method Application of the Stone-Weierstrass theorem to show uniqueness in inverse problems.
result Generalization and simplification of earlier results in inverse problems.
The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures satisfying suitable conditions. In this paper…
Similarity algebra extends algebraic structures with quantitative bounds.
problem Exact algebraic structures with strict axioms.
method Framework for approximate algebraic and Lie structures with -estimates.
result Similarity structures converge to classical algebraic objects as .
Proves quantitative Alexandrov theorem for capillary surfaces.
problem Proving a quantitative version of the Alexandrov theorem for capillary hypersurfaces.
method Quantitative analysis of Montiel-Ros-type argument.
result Generalizes Julin-Niinikoski's result to capillary case.
The paper details local forms of morphisms in colored supermanifolds.
problem Understanding local forms of morphisms in colored supermanifolds.
method Detailed account of -differential calculus and local theorems.
result Detailed insights into local forms of morphisms in colored supermanifolds.
The study examines hypersurfaces close to constant mean curvature and their proximity to spheres.
problem Understanding hypersurfaces close to constant mean curvature and their proximity to spheres.
method Quantitative stability results for hypersurfaces with mean curvature close to a constant.
result Hypersurfaces close to constant mean curvature are closely related to spheres, with quantitative descriptions of proximity.
Stable solution found for manifold topology from boundary data.
problem Determining manifold properties from boundary data and eigenvalues.
method Quantitative stability estimates and unique continuation for the wave operator.
result Eigenvalues and boundary values determine a metric space close to the manifold.
Represents neural networks as solutions to inverse problems in Banach spaces.
problem Understanding the function learned by neural networks.
method Variational framework, representer theorem, polynomial ridge splines.
result Neural networks are solutions to inverse problems in Banach spaces.
New findings on mesh group-planes validate Signature-inverse Theorem under specific conditions.
problem Invalidity of existing inverse theorems for mesh group-planes.
method Classification of three and five point meshes, analysis of joint invariant signatures.
result Valid conditions for the Signature-inverse Theorem in mesh group-planes.
Singular implicit and inverse function theorems. Strong resolution with normally flat centersmath.AG
Building upon ideas of Hironaka, Bierstone-Milman, Malgrange and others we generalize the inverse and implicit function theorem (in differential, analytic and algebraic setting) to sets of functions of larger multiplicities (or ideals). This allows one to describe singularities given by a finite set of generators or by…
In this paper we establish the basic tools to develop the "Calculus" associated with group-valued continuously Pansu differentiable mappings. We develop the technical machinery on which all of our results rely. In particular, the linearization of addends appearing in the Baker-Campbell-Hausdorff formula is one of the m…
Framework combines machine learning and inverse methods to quantify uncertainties in model parameters.
problem Combining aleatoric and epistemic uncertainties in engineered systems modeling.
method Develops a robust filtering step in LUQ to learn useful QoI maps from noisy datasets, iterates over time, and uses sufficiency tests.
result Transforms datasets into distributions for DC-based inversion, improving parameter quantification.
The paper constructs metrics on spheres with families of minimal hypersurfaces.
problem Finding Riemannian metrics on spheres with specific families of minimal hypersurfaces.
method Used Nash-Moser Inverse Function Theorem in the tame maps setting.
result Generalized Guillemin's theorem for Zoll families of minimal hypersurfaces.
We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general…
New integral theorems improve density function estimations.
problem Improving density function estimations.
method Integrals based on cyclic functions and Riemann sums, Fourier integral theorem, Monte Carlo methods, variational approach, Cauchy residue theorem.
result Optimal cyclic functions minimize square integrals, improving density estimations.
Proves a limit on hyperplanes in complex manifolds.
problem Limiting the number of hyperplanes in complex manifolds.
method Effective density theorem for periodic orbits, Margulis functions, restricted projection theorem, equidistribution result.
result Proves a quantitative finiteness theorem for hyperplanes.
The paper sharpens a theorem about surfaces with zero Gaussian curvature.
problem Quantifying the isometric property of surfaces with zero Gaussian curvature.
method Asymptotically sharp quantitative version of a classical theorem using isothermal coordinates.
result An isothermal coordinate map from a Riemannian disc to an Euclidean disc is bi-Lipschitz with a constant of exp(4ε).
This article is about inverse spectral problems for hyperbolic surfaces and in particular how length spectra relate to the geometry of the underlying surface. A quantitative answer is given to the following: how many questions do you need to ask a length spectrum to determine it completely? In answering this, a quantit…
Proves a quantitative index theorem for positive scalar curvature metrics.
problem Studying conjectures and open questions on positive scalar curvature.
method Quantitative relative index theorem and -Lipschitz rigidity theorem.
result Positive answers to Gromov's open questions on scalar curvature.
We construct new examples of Einstein metrics by perturbing the conformal infinity of geometrically finite hyperbolic metrics and by applying the inverse function theorem in suitable weighted Hölder spaces.
A theorem for debiasing machine learning with finite sample guarantees.
problem Calculating confidence intervals for machine learning functionals.
method Debiased machine learning based on bias correction and sample splitting.
result Nonasymptotic debiased machine learning theorem with finite sample guarantees.
Quantitative susceptibility mapping (QSM) is a powerful MRI technique that has shown great potential in quantifying tissue susceptibility in numerous neurological disorders. However, the intrinsic ill-posed dipole inversion problem greatly affects the accuracy of the susceptibility map. We propose QSMGAN: a 3D deep con…
Proves existence of proper solutions for inverse mean curvature flow.
problem Existence of proper solutions for inverse mean curvature flow.
method Proves existence theorem assuming non-degeneracy conditions on isoperimetric profile.
result No curvature assumption in existence theorem.
Stability of the positive mass theorem and torus rigidity theorems under integral curvature boundsmath.DG
The paper analyzes harmonic maps to flat model spaces to prove geometric stability of the positive mass theorem.
problem Geometric stability of the positive mass theorem and related conjectures.
method Analysis of harmonic maps to flat model spaces under integral curvature bounds.
result Upgrading harmonic maps to diffeomorphisms when conditions are met, proving quantitative closeness to model spaces.
Paper analyzes arbitrage in uncertain markets, providing quantitative asset pricing.
problem Dealing with model uncertainty in markets that allow small arbitrage.
method Quantitative analysis of arbitrage, focusing on asset price processes close to martingales.
result Quantitative version of the Fundamental Theorem of Asset Pricing and Super-Replication Theorem.
Study graph products of groups, classifying them up to measure equivalence and rigidity.
problem Classifying graph products of groups up to measure equivalence and rigidity.
method Measure-theoretic and structural properties of von Neumann algebras, rigidity theorems.
result Quantified measure equivalence classification and rigidity theorems for graph products.
One of the key issues in the analysis of machine learning models is to identify the appropriate function space and norm for the model. This is the set of functions endowed with a quantity which can control the approximation and estimation errors by a particular machine learning model. In this paper, we address this iss…
New method disentangles perceptual uncertainty and behavioral costs in partially observable systems.
problem Tackles inverse optimal control for non-linear partially observable systems.
method Probabilistic approach using maximum causal entropy formulations and local linearization.
result Disentangles perceptual factors and behavioral costs in sequential decision-making.
Unified view of monotonicity formulas for inverse mean curvature flow and -capacitary potentials.
problem Understanding monotonicity formulas for various geometric flows and potentials.
method Refined analysis of -capacitary potentials and their level sets.
result Strong convergence of -capacitary potentials to inverse mean curvature flow and curvature varifolds.
We introduce and study generalized -harmonic equations (1.1). Using some ideas and techniques in studying -harmonic functions from [W1] (2007), and in studying nonhomogeneous -harmonic functions on a cocompact set from [W2, (9.1)] (2008), we find an analytic quantity in the generalized -harmonic equatio…
It has been known in that round spheres are the only closed homothetic self-similar solutions to the inverse mean curvature flow and parabolic curvature flows by degree -1 homogeneous functions of principle curvatures in the Euclidean space. In this article, we prove that the round sphere is rigid in much stronger sens…
Stability results for geometric equations in warped product spaces.
problem Geometric partial differential equations in warped product spaces.
method Stability theorem development for level sets of functions.
result Quantitative stability theorems for Serrin's problem and Alexandroff's theorem.
Study geodesic Lie groups' convergence to limits with quantitative estimates.
problem Quantifying convergence rates of geodesic Lie groups to their limits.
method Estimates on the difference between original metrics and asymptotic/tangent metrics.
result Sharpens existing bounds on convergence rates.
The paper identifies and critiques problems with risk matrices using ordinal scales.
problem Problems with risk matrices using ordinal scales.
method Overview of risk assessment process, explanation of fallacies, and suggestions for improvement.
result The paper proposes avoiding risk matrices and using fully quantitative methods instead.