New examples of mean curvature flow converge to minimal surfaces with multiplicity 2.
problem Constructing mean curvature flow examples in closed manifolds.
method Constructing new examples of mean curvature flow with convergence to minimal surfaces with multiplicity 2.
result Mean curvature flow examples converge to minimal surfaces with multiplicity 2.
New example of surface flow converging to a plane with multiplicity 2.
problem Constructing mean curvature flows with specific convergence properties.
method Constructing a new example of a mean curvature flow in R3. result The flow converges to a plane with multiplicity 2 as time approaches infinity.
The paper studies a modified scalar curvature flow and proves convergence to a sphere.
problem Analyzing the convergence of a modified scalar curvature flow.
method Flow of starshaped hypersurfaces with a specific speed function, proving existence and convergence.
result The flow converges exponentially fast to a sphere, except for α<2. The paper proves spectral convergence rates for graph Laplacian to manifold Laplace-Beltrami operator.
problem Spectral convergence of graph Laplacian to manifold Laplace-Beltrami operator.
method Analysis of Dirichlet form convergence and construction of approximate eigenfunctions via manifold heat kernel.
result Proves spectral convergence rates for Gaussian kernelized graph Laplacian.
Proves convergence groups on a 2-sphere are Kleinian groups.
problem Proving convergence groups on a 2-sphere are Kleinian groups.
method Analyzing relatively hyperbolic groups with planar boundaries and applying to various versions of the Cannon conjecture.
result Proves relatively hyperbolic groups with planar boundaries are virtually Kleinian.
Hamiltonian Monte Carlo converges to target distributions under mild conditions.
problem Establishing convergence of Hamiltonian Monte Carlo algorithms.
method Analyzing Lq convergence for Hamiltonian Monte Carlo under mild conditions. result Outputs converge to target distributions under specified conditions.
Study on harmonic forms on K3 surfaces converging to a flat 4D orbifold.
problem Behavior of harmonic 2-forms on K3 surfaces with Ricci-flat metrics.
method Analysis of convergence of harmonic forms to flat 4D orbifold.
result Decomposition of harmonic 2-forms into converging subspaces.
Study convergence and approximations of entropic regularized Wasserstein distances for Gaussian and RKHS measures.
problem Convergence and approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings.
method Analysis of convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings.
result Strictly weaker convergence in 2-Sinkhorn divergence for Gaussian measures compared to exact 2-Wasserstein distance.
In this paper, we give a sufficient condition such that the Ricci flow in R2 exists globally and the flow converges at t=∞ to the flat metric on R2.
Study introduces fractional mass concept for surfaces, proving its convergence.
problem Understanding fractional mass on surfaces.
method Introduces fractional s-mass, proves Γ-convergence and pointwise convergence. result Fractional s-mass converges to (n−2)-dimensional area. New bounds on scalar curvature for metric sequences.
problem Bounding scalar curvature in metric sequences.
method Integral convergence of scalar curvature; point-wise scalar curvature lower bound.
result Limiting metric has scalar curvature lower bound.
We prove that, starting at an initial metric g(0)=e2u0(dx2+dy2) on R2 with bounded scalar curvature and bounded u0, the Ricci flow ∂tg(t)=−Rg(t)g(t) converges to a flat metric on R2.
Polynomial networks converge to Gaussian processes at a rate of O(n^(-1/2)).
problem Understanding the convergence rate of polynomial networks to Gaussian processes.
method Examined one-hidden-layer neural networks with random weights, focusing on polynomial activations and their convergence rate in the 2-Wasserstein metric.
result The rate of convergence for polynomial networks to Gaussian processes is $O(n^{- rac{1}{2}})$.
Paper proves convergence rates for Gaussian kernel ridge regression.
problem Understanding convergence rates for Gaussian kernel ridge regression.
method Establishes polynomial convergence rates for KRR with fixed hyperparameters.
result First polynomial convergence rates for Gaussian kernel ridge regression.
We prove a convergence theorem on the moduli space of constant σ2 metrics for conic 4-spheres. We show that when a numerical condition is convergent to the boundary case, the geometry of conic 4-spheres converges to the boundary case while preserving capacity.
The paper studies the convergence of elastic flows of curves into manifolds, proving smooth convergence under certain conditions.
problem The convergence of elastic flows of curves into manifolds.
method Parabolic estimates and Lojasiewicz-Simon gradient inequality.
result Smooth convergence of the flow to critical points under specific conditions.
The paper constructs non-convergent solutions to Vafa-Witten equations with specific harmonic 2-form limits.
problem Constructing solutions to Vafa-Witten equations with non-zero mass term.
method Constructs divergent sequences of solutions, renormalizes them, and defines harmonic 2-form data sets.
result Defines an 'interesting' harmonic 2-form data set with specific properties.
In this paper, a metric with G2 holonomy and slow rate of convergence to the cone metric is constructed on a ball inside the cone over the flag manifold.
We consider linear slices of the space of Kleinian once-punctured torus groups; a linear slice is obtained by fixing the value of the trace of one of the generators. The linear slice for trace 2 is called the Maskit slice. We will show that if traces converge `horocyclically' to 2 then associated linear slices converge…
This short note aims at (re)proving that the symmetrically normalized graph Laplacian $L=\Id - D^{-1/2}WD^{-1/2}$ (from a graph defined from a Gaussian weighting kernel on a sampled smooth manifold) converges towards the continuous Manifold Laplacian when the sampling become infinitely dense. The convergence rate with …
Improved convergence rates for Stein Variational Gradient Descent in finite-particle settings.
problem Improving convergence rates for Stein Variational Gradient Descent in finite-particle settings.
method Analyzing the time derivative of relative entropy and splitting it into dominant and smaller parts.
result Finite-particle convergence rates of order 1/\sqrt{N} for Kernelized Stein Discrepancy and Wasserstein-2 metrics.
New bounds show diffusion models converge nearly linearly in data dimension.
problem Improving convergence bounds for diffusion models.
method Refined discretization of reverse SDE using stochastic localization.
result Linear convergence in data dimension with logarithmic factors.
Compactness theorems for G2-solitons established with scalar curvature and potential function constraints.
problem Establishing compactness theorems for G2-solitons under specific conditions. method Proved Gromov-Hausdorff convergence and derived epsilon-regularity estimates.
result Smooth convergence of G2-solitons under uniform energy bounds at half the dimension. In this note we give a simple proof for the convergence of stochastic gradient (SGD) methods on μ-convex functions under a (milder than standard) L-smoothness assumption. We show that for carefully chosen stepsizes SGD converges after T iterations as $O\left( LR^2 \exp \bigl[-\fracμ{4L}T\bigr] + \frac{σ^2}{μT} \r…
Paper analyzes convergence of DDPM for general distributions.
problem Theoretical understanding of DDPM's convergence properties remains limited.
method Introduced a relaxed smoothness condition and proved near-optimal convergence rates.
result Established a convergence rate of \( \widetilde{O}\left(\frac{d\min\{d,L^2\}}{T^2}
ight) \) in Kullback-Leibler divergence.
Study on convergence rates for optimal transport with regularization.
problem Convergence analysis of divergence-regularized optimal transport.
method Novel methodology using quantization and martingale couplings.
result Sharp rates for various divergences and transport costs.
In this paper we study the family of embeddings Φt of a compact RCD∗(K,N) space (X,d,m) into L2(X,m) via eigenmaps. Extending part of the classical results by Bérard, Bérard-Besson-Gallot, known for closed Riemannian manifolds, we prove convergence as t↓0 of the rescaled pull-back metrics $Φ_t^*g…
For the class of approximate harmonic maps u∈W1,2(Σ,N) from a closed Riemmanian surface (Σ,g) to a compact Riemannian manifold (N,h), we show that (i) the so-called energy identity holds for weakly convergent approximate harmonic maps {un}:Σ→N, with tension fields τ(un) bounded in the Morrey spa…
Deep neural networks without regularization can achieve consistent estimates with good convergence rates.
problem The necessity of regularization in deep neural networks for consistent estimates.
method Gradient descent on an over-parametrized neural network without regularization, with specific initialization, step size, and number of steps.
result An estimate without regularization is universally consistent and achieves good convergence rates.
Paper closes convergence gap for SGD without replacement.
problem Establishing convergence rate for SGD without replacement.
method Analyzes convergence rates for strongly convex and smooth functions.
result Achieves a rate of O(1/T^2 + n^2/T^3) for quadratic sums.
Study on convergence rate of Q-curvature flow in 6 dimensions.
problem Analyzing the convergence rate of Q-curvature flow in 6 dimensions. method Provided an example of a slowly converging Q6-curvature flow in dimension 6. result The Q-curvature flow in 6 dimensions does not always converge exponentially, unlike in 2 dimensions. Uniform convergence of metrics on surfaces with bounded curvature measures proved.
problem Proving uniform convergence of metrics on Alexandrov surfaces with bounded integral curvature.
method Weak convergence of measures and analytic approximation of metrics.
result Uniform convergence of metrics on Alexandrov surfaces proved.
ADOPT optimizes Adam to converge with any β2 without bounded noise.
problem Non-convergence of Adam optimization algorithm.
method ADOPT removes current gradient from second moment estimate and changes momentum update order.
result ADOPT achieves optimal convergence rate of O(1 / √T) with any β2.
We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature f…
The paper studies curvature flows of star-shaped hypersurfaces and proves convergence to spheres.
problem Analyzing the convergence of a class of anisotropic curvature flows.
method Using new auxiliary functions, the paper studies a class of flows with specific speed and proves convergence under certain conditions.
result The k-convex solution to the flow converges smoothly to a sphere after normalization for specific values of k, α, and β. In this paper, we are interested in the strong convergence properties of the Ninomiya-Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order 1/2. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multil…
The paper analyzes how over-parameterization affects GD convergence in matrix sensing problems.
problem Matrix sensing problem with over-parameterized gradient descent.
method Analyzes symmetric and asymmetric parameterizations, provides lower bounds and convergence rates.
result Over-parameterization slows down GD convergence, but asymmetric parameterization can speed up convergence.
We prove the hypersymplectic flow of simple type on standard torus T4 exists for all time and converges to the standard flat structure modulo diffeomorphisms. This result in particular gives the first example of a cohomogeneity-one G2-Laplacian flow on a compact 7-manifold which exists for all time and…
We prove that a sequence of quasi-Fuchsian representations for which the critical exponent converges to the topological dimension of the boundary of the group (larger than 2), converges up to subsequence and conjugacy to a totally geodesic representation.
The paper studies algebraic integer relations and sequences converging to 4.
problem Investigating algebraic integer relations and convergence of sequences.
method Constructing a generalized Farey graph for the subgroup Gα and analyzing its properties. result A sequence of algebraic integers converges to 4, each corresponding to a non-free group of rank 2.
New method solves root-finding problems with faster convergence.
problem Finite-sum co-coercive equations
method Variance-reduced Krasnoselkii--Mann methods
result Achieves both O(1/k2) and o(1/k2) convergence rates The paper shows how Yang-Mills-Higgs energies converge to the (n−2)-area functional.
problem Understanding the convergence of Yang-Mills-Higgs energies to the (n−2)-area functional. method Analyzing the convergence of critical points of Yang-Mills-Higgs energies to minimal submanifolds and proving Γ-convergence. result Yang-Mills-Higgs energies converge to the (n−2)-area functional as εo0. We study the flow Mt of a smooth, strictly convex hypersurface by its mean curvature in Rn+1. The surface remains smooth and convex, shrinking monotonically until it disappears at a critical time T and point x∗ (which is due to Huisken). This is equivalent to saying that the corresponding rescaled…
Study on prescribing positive curvature with conical singularities on a sphere.
problem Prescribing positive curvature with conical singularities on a sphere.
method Fine analysis of bubble trees and an area identity in the convergence process.
result Criterion for nonexistence in an open region of the prescribing data.
GD converges faster to flatter minima than gradient flow in shallow networks.
problem Understanding the dynamics of gradient descent in shallow linear networks.
method Analyzing the convergence rate and solution of gradient descent in depth-2 linear neural networks.
result GD converges linearly to flatter minima than gradient flow, even with large step sizes.
The study finds counterexamples to curvature estimates for minimizing surfaces.
problem Curvature estimates for minimizing surfaces in metric convergence.
method Constructing sequences of smooth minimizing surfaces in metrics converging to Euclidean.
result Found counterexamples with diverging L2 norm of second fundamental form. We study convergence properties of the full truncation Euler scheme for the Cox-Ingersoll-Ross process in the regime where the boundary point zero is inaccessible. Under some conditions on the model parameters (precisely, when the Feller ratio is greater than three), we establish the strong order 1/2 convergence in $L^…
Paper develops MMOT framework for financial applications with neural acceleration.
problem Financial optimization and calibration under multi-period martingale constraints.
method Theoretical analysis, incremental updates, adaptive sparse grids, hybrid neural-projection solver.
result Neural solver achieves 1597x speedup for real-time applications.