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.
Estimates small eigenvalues for geometrically finite manifolds.
problem Estimating small eigenvalues of Schrödinger operators.
method Geometrically finite manifolds, Riemannian vector bundles.
result Estimates the number of small eigenvalues.
Geometric method improves uncertainty estimation in real-time.
problem Improving uncertainty estimation in machine learning models.
method Geometric distance from training inputs for uncertainty estimation, post-hoc calibration.
result Method yields better uncertainty estimations than existing approaches.
We prove uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We also prove a uniform diameter estimate for collapsing families of twisted Kahler-Einstein met…
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 paper estimates gradients for a weighted parabolic equation under geometric flow.
problem Estimating gradients for a specific parabolic equation on a weighted manifold.
method Obtained space-time gradient estimates through integrating the equation.
result Found corresponding Harnack inequalities through gradient estimates.
Extends gradient estimates for heat equation under Finsler geometric flows.
problem Global gradient estimates for positive solutions to heat equation.
method General compact Finsler CD(−K,N) geometric flow. result Derives Harnack inequality for positive solutions.
This paper proposes a geometric estimator of dependency between a pair of multivariate samples. The proposed estimator of dependency is based on a randomly permuted geometric graph (the minimal spanning tree) over the two multivariate samples. This estimator converges to a quantity that we call the geometric mutual inf…
Study geometric inequalities and boundary estimates for Einstein-type manifolds with boundary.
problem Investigate geometric properties of Einstein-type manifolds with boundary.
method Investigate geometric inequalities and establish boundary estimates.
result Established boundary estimates in terms of eigenvalues and Brown-York mass.
Combines Gaussian process and Geometric Harmonics for better uncertainty estimation.
problem Uncertainty estimation in kernel-based methods.
method Combines Gaussian process and Geometric Harmonics.
result Alternative interpretations of uncertainty and accelerated Bayesian Optimization.
Survey on manifold ends with new heat kernel estimates.
problem Analyzing geometric properties on manifolds with ends.
method Constructing manifolds with ends and analyzing their heat kernel estimates.
result Found manifolds with ends that have different heat kernel estimates.
In general, gradient estimates are very important and necessary for deriving convergence results in different geometric flows, and most of them are obtained by analytic methods. In this paper, we will apply a stochastic approach to systematically give gradient estimates for some important geometric quantities under the…
This paper introduces online algorithms to estimate robust geometric median in large data streams.
problem Detecting outliers in large data sets using robust statistical measures.
method Online stochastic Newton methods for estimating the geometric median.
result Rates of convergence for online estimation of the geometric median.
The paper proposes estimators for bid-ask spreads with and without serial dependence.
problem Estimating bid-ask spreads in financial markets with and without serial dependence.
method The authors propose moment-based estimators for bid-ask spreads, considering both geometric Brownian motion and geometric fractional Brownian motion for price dynamics, and Ornstein-Uhlenbeck process for microstructure noise.
result The estimators are consistent and asymptotically normal, and perform well compared to existing approaches on simulated data.
New approach for uniform estimates in complex equations.
problem Uniform estimates for solutions to complex Monge-Ampere equations.
method Efficient new approach to uniform estimates.
result Efficient method for uniform estimates in geometric PDEs.
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.
Study geometric inequalities for quasi-Einstein manifolds using new formulas.
problem Investigate geometric inequalities on quasi-Einstein manifolds.
method Use generalized Reilly's formulas and establish new boundary estimates and isoperimetric inequalities.
result Present a Heintze-Karcher type inequality for compact quasi-Einstein manifolds.
In the present paper, we obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation under general geometric flow on complete noncompact manifolds.
New bounds for geometric flows of Hermitian metrics established.
problem Regularity of geometric flows of Hermitian metrics.
method Establishing a C1 a priori bound for smooth curves of Hermitian metrics. result New regularity result for Hermitian curvature flows, including the second Chern-Ricci flow.
The paper provides gradient estimates for a parabolic equation under Finsler geometric flows.
problem Gradient estimates for a general parabolic equation under compact Finsler CD(−K,N) geometric flows. method Presented Shi-type and Hamilton-type gradient estimates.
result Demonstrates the possibility of removing stricter derivative bounds imposed by Finsler curvature conditions.
Study on electrostatic systems with boundary, proving new geometric inequalities.
problem Electrostatic systems with boundary in higher dimensions.
method Investigation of electrostatic systems on compact manifolds with boundary, establishing new geometric properties.
result Proved sharp boundary estimates and isoperimetric-type inequalities for electrostatic manifolds.
Geometric focusing affects dispersive estimates for Schrödinger and wave equations.
problem Long-time decay rate in dispersive estimates for Schrödinger and wave equations on non-trapping asymptotically conic manifolds and exact metric cones.
method Classifying the long-time decay rate in dispersive estimates for the Schrödinger and wave equations on non-trapping asymptotically conic manifolds and exact metric cones in terms of the intensity of geometric focusing.
result Each multiplicity of conjugate points within distance π on Y = ∂X0 leads to a |t|1/2-loss in the long-time decay order and a half-order shift in the regularity index in the dispersive estimate for the Schrödinger equation.
Develops local elliptic regularity for geometrically-natural operators with low regularity coefficients.
problem Local elliptic regularity for operators with low regularity coefficients in Sobolev-type spaces.
method Rescaling estimates and multiplication results for function spaces.
result Unified set of interior estimates and regularity inference for operators with Sobolev-type coefficients.
Extends heat kernel estimates for super Ricci flow.
problem Heat kernel estimates for super Ricci flow.
method Generalizes Bamler-Zhang's geometric analysis to super Ricci flow.
result Obtains Gaussian heat kernel estimates for super Ricci flow.
This paper presents a unified geometric framework for the statistical analysis of a general ill-posed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix estimation, and noisy matrix completion. We propose computationally feasible conv…
We estimate from below by geometric data the eigenvalues of the periodic Sturm-Liouville operator −4d2/ds2+κ2(s) with potential given by the curvature of a closed curve.
Optimal geometric estimates for Kähler manifolds with bounded Nash entropy
problem Optimal geometric estimates for compact Kähler manifolds
method Proving Sobolev-type inequality and local volume noncollapsing with optimal exponents
result Uniformly bounded q-Nash entropy Study geometric properties and spectral estimates on warped products.
problem Investigate Ricci curvature and spectral estimates in warped products.
method Establish integral inequalities and sufficient conditions for geometric properties.
result Sufficient conditions for intersection of warped products with totally geodesic hypersurfaces.
New approach combines geometric and probabilistic methods to estimate manifold dimension in high-dimensional data.
problem Estimating the dimension of manifolds in high-dimensional data.
method Combines a modified box-counting algorithm (geometric) and a new probabilistic method (nearest neighbor distance analysis).
result The combined method is robust, fast, and effective in estimating manifold dimension.
In this paper, we consider a manifold evolving by a general geometric flow and study parabolic equation \[ (Δ-q(x,t)-\partial_t)u(x,t)=A(u(x,t)),\quad (x,t)\in M\times [0,T]. \] We establish space-time gradient estimates for positive solutions and elliptic type gradient estimates for bounded positive solutions of this …
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-weighted Ricci curvature bounds. result Proves eigenvalue estimates for Steklov and ABP inequalities on weighted manifolds.
Sharp ABP estimate on metric spaces via optimal transport.
problem Sharp ABP estimate on metric measure spaces.
method Optimal transport theory.
result Established a sharp ABP estimate on metric measure spaces.
We study the problem of supervised learning for both binary and multiclass classification from a unified geometric perspective. In particular, we propose a geometric regularization technique to find the submanifold corresponding to a robust estimator of the class probability P(y∣x). The regularization term meas…
Paper finds convexity in translating solitons for concave flows.
problem Understanding convexity in translating solitons for concave extrinsic flows.
method Analyzes convexity estimates for translating solitons evolving under concave functions in Rn+1. result Establishes convexity estimates for translating solitons of concave extrinsic geometric flows.
Paper proposes a robust method for federated ICA with geometric median aggregation.
problem Federated ICA with permutation ambiguity in client estimations.
method Geometric median aggregation with k-means clustering to resolve permutation ambiguity.
result The method provably remains effective in highly heterogeneous scenarios.
Study geometric structures in transfer learning to avoid negative transfer.
problem Understanding information-theoretic limits of transfer learning without exploiting domain geometry.
method Integrates geometric structure into linear regression models, using Gram matrices of source and target domains.
result Proposes an interpolation estimator that matches minimax lower bound and outperforms existing methods.
Estimating dimension from sparse random geometric graphs.
problem Estimating the dimension of the underlying space from a random geometric graph.
method An estimator of dimension is derived using the adjacency matrix of the graph, under specific conditions on the density and threshold.
result An estimator converges to the true dimension with high probability under certain conditions.
Improved private geometric median estimation with nearly-linear time complexity.
problem Estimating the geometric median of a dataset while maintaining privacy.
method Improved algorithm using subsampling and geometric aggregation, achieving nearly-linear runtime.
result Achieves the same approximation quality as previous methods but with nearly-linear runtime.
We explore the connection between two problems that have arisen independently in the signal processing and related fields: the estimation of the geometric mean of a set of symmetric positive definite (SPD) matrices and their approximate joint diagonalization (AJD). Today there is a considerable interest in estimating t…
Study geometric and analytical properties of ρ-Einstein solitons.
problem Characterize geometric and analytical features of ρ-Einstein solitons. method Analyze the spectrum of the drifted Laplacian operator and prove volume growth estimates.
result Establish new volume growth estimates for geodesic balls of complete noncompact ρ-Einstein solitons. The paper introduces a geometric flow for Lagrangian submanifolds that preserves Hamiltonian isotopy.
problem Finding stationary solutions for Hamiltonian stationary Lagrangian submanifolds.
method Introducing a geometric flow that is a gradient flow for volume and corresponds to a fourth order strictly parabolic scalar equation.
result Established short-time existence, uniqueness, and higher order estimates for compact initial Lagrangian immersions with uniformly bounded second fundamental forms.
Study eigenvalues on quaternion-Kähler manifolds with geometric bounds.
problem Estimating eigenvalues on quaternion-Kähler manifolds.
method Lower bounds derived from modulus of continuity estimates for heat equation solutions and Laplace comparison theorem.
result Established bounds for first nonzero eigenvalues in terms of dimension, diameter, and scalar curvature.
Derives estimates for geometric elliptic equations on complex manifolds.
problem Estimating solutions of geometric elliptic equations on complex manifolds.
method Derives a priori real Hessian estimates independent of the right-hand side.
result Establishes optimal C1,1 regularity of geometric envelopes. Let (M,g(t)), t∈[0,T) be a closed Riemannian n-manifold whose Riemannian metric g(t) evolves by the geometric flow ∂t∂gij=−2Sij, where Sij(t) is a symmetric two-tensor on (M,g(t)). We discuss differential Harnack estimates for positive solution to the porous medium …
Let M be a closed Riemannian manifold with a family of Riemannian metrics gij(t) evolving by geometric flow ∂tgij=−2Sij, where Sij(t) is a family of smooth symmetric two-tensors on M. In this paper we derive differential Harnack estimates for positive solutions to the nonlinear heat …
We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We focus on determining when a link is hyperbolic, estimating its volume, and boundi…
Study proves interaction of three impulsive gravitational waves, showing local solution and Lipschitz continuity.
problem Interaction of three impulsive gravitational waves in Einstein vacuum equations.
method Geometric estimates and wave estimates to prove local solution and continuity.
result Local solution to Einstein vacuum equations with three impulsive gravitational waves, Lipschitz continuity.
Study on compact Quasi-Einstein manifolds yields diameter estimates and Hitchin-Thorpe inequality conditions.
problem Estimating diameters and verifying Hitchin-Thorpe inequality for compact Quasi-Einstein manifolds.
method Derive geometric estimates relating potential function oscillation to manifold diameter; derive lower bounds for diameter.
result Diameter conditions ensure compact Quasi-Einstein manifolds satisfy Hitchin-Thorpe inequality in dimension four.