Clarifies definition of polarized canonical radius in Kahler Ricci flow.
problem Unclear definition of polarized canonical radius in Kahler Ricci flow.
method Clarification of the definition.
result Clarified definition of polarized canonical radius.
Given a positive function u∈W1,n, we define its John-Nirenberg radius at point x to be the supreme of the radius such that ∫Bt∣∇logu∣n<ε0n when n>2, and ∫Bt∣∇u∣2<ε02 when n=2. We will show that for a collapsing sequence in a fixed conformal class under some curvature c…
Random translation surfaces converge to a Poisson plane as genus grows.
problem Understanding the geometric behavior of high genus translation surfaces.
method Proving convergence of random translation surfaces to a Poisson plane using statistical local geometric properties.
result The radius-r neighborhood of a random point in an MSV-distributed random translation surface converges in distribution to the radius r neighborhood of the root in a Poisson translation plane. Local limit theorem for random walks on hyperbolic groups with parabolic subgroups.
problem Analyzing the behavior of random walks on relatively hyperbolic groups.
method Study of convergent random walks with finite derivative of Green function at spectral radius.
result Proves a local limit theorem for the probability of returning to the origin.
The paper analyzes how quantum PageRank changes with small perturbations.
problem Estimating sensitivity of quantum PageRank to small changes.
method Finite dimensional perturbation theory to estimate changes and bounds.
result Estimation of lower bound of convergence radius and error bounds.
Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using Ck,α convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein m…
In this paper, we mainly study the compactness and local structure of immersing surfaces in Rn with local uniform bounded area and small total curvature ∫Σ∩B1(0)∣A∣2. A key ingredient is a new quantity which we call isothermal radius. Using the estimate of the isothermal radius we establish a…
Study circular foliations and shear-radius coordinates on hyperbolic cone surfaces.
problem Characterize Teichmüller spaces of hyperbolic cone surfaces.
method Construct circular foliations and shear-radius coordinates on Teichmüller spaces of hyperbolic cone surfaces.
result Shear-radius coordinates provide global coordinates on Teichmüller spaces and converge to specific metrics.
Polyak step size GD reaches final radius of convergence after log iterations.
problem Statistical and computational complexities of Polyak step size GD.
method Generalized smoothness and Lojasiewicz conditions, stability of gradients.
result Polyak step size GD reaches final statistical radius of convergence after logarithmic number of iterations.
The paper interprets learned step sizes in deep-unfolded gradient descent.
problem Intuitive interpretation of learned non-constant step sizes in deep-unfolded gradient descent.
method Theoretical analysis and optimization of spectral radius.
result Chebyshev steps achieve the lower bound of convergence rate for first-order methods.
First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space under certain condition for the radius function. Next, we prove that the …
Paper proves unique tangent flow at infinity for entropy-limited curve shortening.
problem Proving uniqueness of tangent flows for finite-entropy curve shortening.
method Rescaled backward convergence to a line, entropy analysis, and geometric properties.
result Ancient smooth curve shortening flow has a unique tangent flow at infinity.
Graph Laplacian approximates manifold eigenvalues with controlled curvature bounds.
problem Approximating eigenvalues of Laplace-Beltrami on manifolds with bounded Ricci curvature.
method Graph discretization of Riemannian manifolds with (ε,ρ)-approximation, proving eigenvalue convergence. result Graph Laplacian eigenvalues converge uniformly to manifold Laplacian eigenvalues as parameters approach zero.
The paper studies a flow of spacelike curves in a Lorentz-Minkowski plane, showing convergence to a constant function.
problem Evolution of spacelike graphic curves in Lorentz-Minkowski plane.
method Anisotropic inverse mean curvature flow with vanishing Neumann boundary condition.
result The evolving curves converge to a constant function as time tends to infinity.
BMM algorithm improves convergence for nonconvex optimization problems.
problem Constrained nonsmooth nonconvex optimization problems.
method Block majorization-minimization with diminishing radius.
result Improved convergence rate for nonconvex optimization problems.
Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(M_i,g_i(t), O_i)} such that t in [0,T] are solutions to the Ricci flow and g_i(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then t…
We show that there exist hyperbolic knots in the 3-sphere such that the set of points of large injectivity radius in the complement take up the bulk of the volume. More precisely, given a finite volume hyperbolic manifold, for any bound R>0 on injectivity radius, consider the set of points with injectivity radius at le…
Formula for European option pricing under jump diffusion model.
problem Option pricing under complex stochastic processes.
method Infinite series of Black-Scholes terms for Levy-driven processes.
result Series solution converges with a radius of convergence.
The paper studies a flow of spacelike surfaces in Lorentz-Minkowski space, proving convergence to a hyperbolic plane.
problem Evolution of spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Anisotropic inverse mean curvature flow with Neumann boundary condition.
result The flow converges to a hyperbolic plane as time tends to infinity.
TRSVR combines SVRG with trust-region for faster optimization.
problem Unconstrained nonconvex optimization problems.
method Adaptive stochastic trust-region method with variance reduction.
result Converges to first-order stationary points with SVRG.
Spheres' spectral structure converges to Gaussian space's as dimensions grow.
problem Understanding spectral convergence between high-dimensional spheres and Gaussian spaces.
method Proving spectral convergence using projections and eigenvalues.
result Spectral structure on high-dimensional spheres converges to Gaussian space's as dimensions increase.
The Hidden Markov Model (HMM) is one of the mainstays of statistical modeling of discrete time series, with applications including speech recognition, computational biology, computer vision and econometrics. Estimating an HMM from its observation process is often addressed via the Baum-Welch algorithm, which is known t…
The paper studies how certain spacelike surfaces evolve over time in a specific space.
problem Evolution of spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Inverse mean curvature flow with vanishing Neumann boundary condition.
result The evolving surfaces converge to a hyperbolic plane as time goes to infinity.
Optimal financial strategies minimize risk under uncertain models.
problem Maximizing utility in financial markets with model uncertainty.
method Optimized strategies converge to those with minimal norm as uncertainty increases.
result Optimal strategies with minimal norm emerge as uncertainty grows.
Ancient Ricci flows on compact spaces converge to solitons.
problem Understanding long-time behavior of Ricci flows on compact spaces.
method Proving precompactness of invariant metrics and analyzing blow-down sequences.
result Ancient homogeneous Ricci flows on compact manifolds converge to gradient shrinking solitons.
We study a form of cyclic pursuit on Riemannian manifolds with positive injectivity radius. We conjecture that on a compact manifold, the piecewise geodesic loop formed by connecting consecutive pursuit agents either collapses in finite time or converges to a closed geodesic. The main result is that this conjecture is …
We construct hyperbolic integer homology 3-spheres where the injectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic 3-manifolds which Benjamini-Schramm converge to H^3 whose normalized Ray-Singer analytic torsions do not converge to …
New similarity measure for covariate shift improves nonparametric regression rates.
problem Improving nonparametric regression under covariate shift.
method Introducing a new similarity measure based on probability ratios.
result Shows a sharper rate of convergence compared to transfer exponent.
In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold M: 1) the convexity radius of p, $\operatorname{conv}(p)\ge \min\{\frac{1}{2}\operatorname{inj}(p),\operatorname{foc}(B_{\operatorname…
Positive injectivity radius for manifolds with Lie structure at infinity.
problem Injectivity radius positivity for manifolds with specific boundary conditions.
method Lie groupoids to prove injectivity radius positivity.
result Injectivity radius is positive for manifolds with Lie structure at infinity.
Study on critical faces convergence in a Poisson point process.
problem Convergence of point processes associated with critical faces in a Čech filtration.
method Established convergence in M0-topology for critical faces above vanishing threshold. result Obtained limit theorems for positive and negative critical faces.
The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with focal points but no conjugate points. The approach is suggested by a characteriz…
Develops an online Gaussian process method that maintains convergence guarantees without sample complexity issues.
problem The computational intractability of Gaussian processes with streaming data.
method Parsimonious Online Gaussian Processes (POG) that maintains asymptotic consistency with bounded memory.
result POG preserves convergence guarantees to the population posterior with finite memory, even for constant error radius.
We establish a C1,α compactness theorem for the metrics with bounded self - dual Weyl tensor and Scalar curvature. The key step is to estimate the C1,α harmonic radius, where we use the blow up analysis as in \cite{Anderson90}. The result is motivated by, and may be applied to the Calabi flow on complex surfa…
Uniform curvature bounds for regularized metrics with bounds on Ricci tensor and injectivity radius.
problem Bounding curvature of regularized metrics with constraints on Ricci tensor and injectivity radius.
method Mollification of riemannian metrics, uniform W2,p-harmonic radius bounds, Ricci tensor bounds, injectivity radius bounds. result Uniform estimate on the change of sectional curvature for regularized metrics.
This paper analyzes how periodic and soft target updates stabilize linear Q-learning.
problem Theoretical explanation of stabilization mechanisms for linear Q-learning.
method Exact analysis using switched linear system dynamics and the joint spectral radius.
result Periodic and soft target updates can guarantee convergence to the exact projected Q-Bellman solution under specific conditions.
Upper bound for conjugate radius in open manifolds with scalar curvature and spectrum constraints.
problem Bounding the conjugate radius of open manifolds with specific curvature and spectrum conditions.
method Established an upper bound using scalar curvature and bottom-of-spectrum constraints.
result For certain conditions, the conjugate radius is no more than π.
Proves upper bound on filling radius for manifolds with positive scalar curvature.
problem Bounding the filling radius of manifolds with positive scalar curvature.
method Quantitative operator K-theory and index theory.
result Proves a quantitative upper bound on the filling radius.
Compact theorem for minimal surfaces with lower injectivity radius.
problem Proving compactness of minimal surfaces with lower injectivity radius.
method Variant of Choi--Schoen compactness theorem, focusing on injectivity radius.
result Proved compactness theorem for minimal surfaces.
ELU algorithm improves on EM for over-specified Gaussian mixtures.
problem Slow convergence of EM in over-specified Gaussian mixtures.
method Developed ELU algorithm for two-component mixtures, combining exponential location update and gradient descent.
result ELU converges to final statistical radius after logarithmic iterations, resolving open question.
Injectivity radius on Stiefel manifold is π.
problem Determining the injectivity radius of the compact Stiefel manifold.
method Utilized the property of geodesics being space curves of constant Frenet curvatures.
result The injectivity radius on the Stiefel manifold under the Euclidean metric is π.
Lower bound on boundary injectivity radius for specific tubes.
problem Estimating the boundary injectivity radius of Margulis tubes.
method Using curvature bounds to derive a lower bound.
result A lower bound on the boundary injectivity radius is provided.
Study gives bounds on filling radius for Riemannian manifolds.
problem Finding bounds on the filling radius of Riemannian manifolds.
method Curvature-dependent bounds for all closed manifolds and upper bounds for submersion and submetry cases.
result Upper and lower bounds on the filling radius for specific types of manifolds.
The paper investigates learning conditional distributions on multi-dimensional spaces using clustering and neural networks.
problem Learning conditional distributions on multi-dimensional spaces with varying dimensions.
method The approach involves clustering data near varying query points in the feature space to create empirical measures in the target space using two clustering schemes: fixed-radius ball and nearest neighbors. The convergence rates of both methods are analyzed, and the nearest neighbors method is incorporated into neural network training.
result The empirical analysis shows that the nearest neighbors method has better performance in practice and can adapt to a suitable level of Lipschitz continuity locally.
This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the…
The paper proves estimates and theorems for Kähler manifolds.
problem Curvature conditions on Kähler manifolds.
method Volume comparison and rigidity theorems.
result Conjugate radius estimates for Kähler manifolds.
Graphons connect graph structures to manifold properties.
problem Interpolating between graphs and manifolds.
method Graph-to-graphon and graphon-to-manifold convergence.
result Established monotonicity inequality linking combinatorial and geometric parameters.
Improved computational complexity in statistical models using second-order information.
problem Polynomial convergence of gradient descent in singular statistical models.
method Normalized Gradient Descent (NormGD) algorithm with second-order information.
result NormGD reaches final statistical radius in logarithmic iterations of n.