The paper analyzes deep neural networks using control theory to set a time limit for their convergence.
problem Understanding the finite-time convergence of deep neural networks.
method Lyapunov based analysis of the loss function, control theory framework, finite-time control of non-linear systems.
result A priori guarantees of finite-time convergence for deep neural networks are provided.
Optimal estimates derived for residual networks' generalization error.
problem Estimating the generalization error of residual networks.
method Derives optimal a priori estimates using a weighted path norm.
result Optimal error estimates are comparable to Monte Carlo error rates.
PAC learning sample complexity is decidable with finite support bounds.
problem Determining the exact sample complexity for PAC learning concepts.
method Observation and proof of decidability with a-priori bounds.
result Sample complexity can be exactly determined for various concepts with finite support bounds.
Estimates for metrics with constant Chern scalar curvature on complex manifolds.
problem Finding metrics with constant Chern scalar curvature on complex manifolds.
method Proving a priori estimates conditional on an upper bound on entropy.
result Extending a recent result by Chen-Cheng in the Kähler setting.
Study shows how certain expanding spacetimes can collapse into flat or Kasner spacetimes.
problem Understanding the collapse of expanding vacuum spacetimes.
method Analysis of spacetimes with CMC foliations and scale invariant a priori bounds.
result Arbitrarily large future time intervals can be modelled by flat or Kasner spacetimes.
New method for complex Monge-Ampère equations on Kähler manifolds.
problem Degenerate complex Monge-Ampère equations on complex manifolds.
method New approach relying on compactness and envelopes properties of quasi-plurisubharmonic functions.
result New and efficient proofs of fundamental results in Kähler geometry.
Bound on singular points for area-minimizing surfaces.
problem Understanding singular points on area-minimizing surfaces.
method Provided a bound on the measure of singular points in terms of the boundary geometry.
result An a priori bound on the (n-7)-dimensional measure of the singular set.
Estimates for neural network risk nearly match Monte Carlo error rates.
problem Understanding the performance of two-layer neural networks.
method Established a priori estimates for the population risk of two-layer neural networks.
result The new estimates are nearly optimal and depend only on function norms, not model parameters.
New method removes scalar curvature assumption in Ricci flow smoothing.
problem Uniform bounds on scalar curvature and other factors for Ricci flow.
method Quantitative short-time existence of Ricci flow without scalar curvature assumption.
result Ricci flow smoothing for measure space limits, Gromov-Hausdorff compactness, and topological rigidity results.
New bound shows KFCV's concentration is due to stability, not bias or variance.
problem Analyzing the concentration of k-fold cross-validation estimates.
method Derive an exponential tail inequality for the concentration of a function of random variables, then use it to analyze KFCV.
result The concentration of KFCV is due to the stability of the learning rule and the number of folds, not bias or variance.
New method solves complex Monge-Ampère equations on hermitian manifolds.
problem Solving degenerate complex Monge-Ampère equations on hermitian manifolds.
method New approach using compactness and envelopes properties of quasi-plurisubharmonic functions.
result New relative a priori estimates and existence results for degenerate complex Monge-Ampère equations.
Paper analyzes DRM for solving high-dimensional elliptic PDEs with generalization bounds.
problem Analyzing generalization error of neural network methods for high-dimensional PDEs.
method Developed a new solution theory for spectral Barron space and derived generalization error bounds.
result Generalization error bounds are independent of dimension and solutions lie in spectral Barron space.
Under the assumption of the uniform local Sobolev inequality, it is proved that Riemannian metrics with an absolute Ricci curvature bound and a small Riemannian curvature integral bound can be smoothed to having a sectional curvature bound. This partly extends previous a priori estimates of Ye Li (J. Geom. Anal. 17 (20…
Estimates generalization error for two-layer ReLU NNs through minimum norm solutions.
problem Estimating generalization error for two-layer ReLU NNs trained by mean squared error.
method Uses minimum norm solutions and Neural Tangent Kernel (NTK) regime to derive generalization error bounds.
result Derives an a priori generalization error bound for two-layer ReLU NNs without requiring exponentially large number of neurons.
Let X be a compact Kähler manifold and $\om$ a smooth closed form of bidegree (1,1) which is nonnegative and big. We study the classes ${\mathcal E}_χ(X,\om)$ of $\om$-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight χ has fast growth at infinity, the corresponding functions are …
Study on harmonic functions in RCD spaces, focusing on singular points and vanishing gradients.
problem Behavior of harmonic functions at singular points of RCD spaces.
method Analysis of tangent cones and modulus of continuity.
result Gradient of harmonic functions vanishes at certain singular points.
Study negative scalar curvature metrics with positive boundary mean curvature.
problem Bounding conformal metrics with specific curvature properties.
method Analyzing Riemannian manifolds with boundary conditions.
result A priori boundedness of metrics in specific cases.
New bounds adaptively control spectral complexity of trained Transformers.
problem Understanding why Transformers generalize well in machine learning.
method Spectrum-adaptive post hoc generalization bounds for multi-layer Transformers.
result Bounds adaptively trade off spectral complexity against dimension and depth factors.
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.
Derives Hessian estimates for Lagrangian mean curvature equation.
problem Lagrangian mean curvature equation with supercritical phase and bounded second derivatives.
method Derives a priori interior Hessian estimates.
result Hessian estimates for Lagrangian mean curvature equation.
New bounds for nearly-linear networks without training.
problem Generalization of neural networks close to linearity.
method Perturbation of linear networks to derive bounds.
result First non-vacuous bounds for neural nets.
Paper constructs non-symmetric collapsing spacetimes without symmetries.
problem Forming non-symmetric collapsing spacetimes in vacuum.
method Modified Christodoulou's a priori estimates and gluing construction.
result Past geodesic completeness and asymptotic Minkowski space.
Solves Dirichlet problem for fully nonlinear equations on Hermitian manifolds.
problem Solving Dirichlet problem for fully nonlinear equations on Hermitian manifolds.
method Derived C2 estimates and gradient estimates for solutions. result Solved Dirichlet problem with admissible subsolutions in some cases.
We study critical Riemannian 4-manifolds with a lower bound on Ricci curvature, but no a priori analytic constraints such as on Sobolev constants. We derive elliptic-type estimates for the local curvature radius, which itself controls sectional curvature. The primary method is construction of blow-ups of degenerating m…
We give the first part of a proof of Thurston's Ending Lamination conjecture. In this part we show how to construct from the end invariants of a Kleinian surface group a ``Lipschitz model'' for the thick part of the corresponding hyperbolic manifold. This enables us to describe the topological structure of the thick pa…
In this paper we present a proof of a Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds. A key p oint is the simple geometric nature of the constant in the a priori estimate of this maximum principle. In particular, this maximum principle can be applied to manifolds with Ricci curv…
In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a bounded domain (with smooth boundary) in a given complete (not compact a priori) Riemannian manifold with Ricci bounded below . For this, we use test functions for the Rayleigh quotient subordinated to a family of open se…
In the product space H^n \times R; we obtain uniform a priori C^0 horizontal length estimates, uniform a priori C^1 boundary gradient estimates, as well as uniform modulus of continuity, for a class of horizontal minimal equations. In two independent variables, we derive a certain uniform global a priori C^1 estimates …
Global existence and convergence of pluriclosed flow on Oeljeklaus-Toma manifolds.
problem Global existence and convergence of pluriclosed flow on specific complex manifolds.
method Established global existence with arbitrary initial data and Gromov-Hausdorff convergence of blowdown limits.
result Gromov-Hausdorff convergence of blowdown limits to a torus under conjectural bounds.
Greedy algorithms which use only function evaluations are applied to convex optimization in a general Banach space X. Along with algorithms that use exact evaluations, algorithms with approximate evaluations are treated. A priori upper bounds for the convergence rate of the proposed algorithms are given. These bounds…
New bounds on manifold Betti numbers derived from semigroup norms.
problem Estimating the first Betti number of compact Riemannian manifolds.
method Birman-Schwinger principle and Schatten norm estimates for semigroup differences, without ultracontractivity assumptions.
result Explicit bounds on Betti numbers depend on Ricci tensor norms.
Study complex Monge-Ampère operator on weighted pluricomplex energy classes.
problem Characterize the range of the Complex Monge-Ampère Operator on weighted pluricomplex energy classes.
method Characterizations and a priori estimates on sub-level sets of solutions.
result A non-negative Borel measure is the Monge-Ampère of a unique function in \(\mathcal E_χ\) if and only if \(χ(\mathcal E_χ) \subset L^1(dμ)\).
We find sharp bounds for the norm inequality on a Pseudo-hermitian manifold, where the L^2 norm of all second derivatives of the function involving horizontal derivatives is controlled by the L^2 norm of the sub-Laplacian. Perturbation allows us to get a-priori bounds for solutions to sub-elliptic PDE in non-divergence…
We prove a priori bounds for the trace of the second fundamental form of a C4 isometric embedding into Rn+1 of a metric g of non-negative sectional curvature on Sn, in terms of the scalar curvature, and the diameter of g. These estimates give a bound on the extrinsic geometry in terms of intrinsic quanti…
We determine the Hausdorff limit-set of the Euclidean hypersurfaces with large λ1 or small extrinsic radius. The result depends on the Lp norm of the curvature that is assumed to be bounded a priori, with a critical behaviour for p equal to the dimension minus 1.
In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form H and extend the result to n-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimension…
Geodesics between Kähler potentials are C^{1,1} regular.
problem Regularity of geodesics in Kähler metrics.
method Interior real Hessian bound for complex Monge-Ampere equation.
result Geodesics are C^{1,1} regular.
We give a characterization of critical points that allows us to define a metric invariant on all Riemannian manifolds M with a lower sectional curvature bound and an upper radius bound. We show there is a uniform upper volume bound for all such manifolds with an upper bound on this invariant. We generalize results by…
Establishes 4D regularity for certain metric spaces.
problem Noncollapsed sequences of metrics with Ricci tensor bounds.
method A priori L2 curvature estimates.
result Diffeomorphism finiteness and rigidity theorems.
In this paper we describe the well studied process of renormalization of quadratic polynomials from the point of view of their natural extensions. In particular, we describe the topology of the inverse limit of infinitely renormalizable quadratic polynomials and prove that when they satisfy a-priori bounds, the topolog…
Derives spacetime regularity under specific curvature conditions.
problem Ensuring smoothness in spacetime models with given curvature constraints.
method General regularity estimate for 4-d spacetimes, using Ricci curvature and Lie derivatives.
result Establishes conditions for smoothness in spacetime models.
Uniform bounds found for Sierpinski carpet hyperbolic components.
problem Bounding hyperbolic components of Sierpinski carpet type.
method Establishing uniform a priori bounds and analyzing quadratic-like restrictions.
result Sierpinski carpet hyperbolic components of disjoint type are bounded.
The paper shows how almost isoperimetric domains are close to spheres.
problem Understanding the geometry of almost isoperimetric domains.
method Analyzing finite perimeter subsets with small isoperimetric deficit and applying integral curvature bounds.
result Finite perimeter subsets with small isoperimetric deficit are close to spheres up to a small measure.
New PINNs method improves accuracy in computing Mean Escape Time from bounded domains.
problem Computing Mean Escape Time from bounded domains with high accuracy.
method Boundary-adapted Physics-Informed Neural Networks (PINNs) with exact Dirichlet boundary enforcement.
result Derivation of H2(Ω) a priori error bounds for PINNs with normalized distance approximations. In this paper, we are concerned with the regularity of noncollapsed Riemannian manifolds (Mn,g) with bounded Ricci curvature, as well as their Gromov-Hausdorff limit spaces (Mjn,dj)⟶dGH(X,d), where dj denotes the Riemannian distance. Our main result is a solution to the codimen…
Novel graph theory for neural networks improves understanding of their structure and performance.
problem Understanding the structural benefits and generalization power of neural networks.
method Developed a novel graph theoretical formulation and extended error analysis for neural networks.
result Similar a priori estimates can be obtained for neural networks under certain conditions, independent of input dimension.
We solve a nonconvex optimization problem to find approximate joint triangularizers of noisy matrices.
problem Finding approximate joint triangularizers of noisy matrices.
method Assuming input matrices are perturbations of noise-free, simultaneously diagonalizable matrices, we provide perturbation bounds and solve a nonconvex optimization problem.
result It is possible to find a good initial triangularizer such that the solution obtained by any local descent-type algorithm has certain global guarantees.
Estimates for special Lagrangian curvature equations in critical and convex cases.
problem Interior estimates for special Lagrangian curvature equations.
method Establishes a priori interior curvature and gradient estimates.
result Proves interior curvature and gradient estimates for special Lagrangian curvature equations.