Under weak regularity assumptions, only, we develop a fully geometric theory of vacuum Einstein spacetimes with T2 symmetry, establish the global well-posedness of the initial value problem for Einstein's field equations, and investigate the global causal structure of the constructed spacetimes. Our weak regularity ass…
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48 results for “1}-regular initial sets”
Solves initial value problem for harmonic maps on specific manifolds.
problem Initial value problem for harmonic maps on cohomogeneity one manifolds.
method Setup and solve the initial value problem using equivariant harmonic maps and regular-singular systems.
result Local existence of harmonic maps in a neighborhood of singular orbits.
We consider the Einstein-Maxwell-fluid constraint equations, and make use of the conformal method to construct and parametrize constant-mean-curvature hyperboloidal initial data sets that satisfy the shear-free condition. This condition is known to be necessary in order that a spacetime development admit a regular conf…
Study proves smooth solutions for fractional mean curvature flow within short time.
problem Short-time existence of smooth solutions for fractional mean curvature flow.
method Established using short-time existence theorem for bounded, C^{1,1}-regular initial sets.
result Smooth solutions exist for both fractional mean curvature flow and volume preserving flow.
New initialization methods speed up Sinkhorn algorithm for OT problems.
problem Improving runtime of the Sinkhorn algorithm for optimal transport problems.
method Data-dependent initializers for Sinkhorn algorithm, based on closed-form solutions for specific settings.
result Data-dependent initializers result in dramatic speed-ups without affecting differentiability.
Study of Willmore energy on sphere sublevel sets and flow singularities.
problem Understanding the Willmore energy landscape and singularities of the Willmore flow.
method Gluing different instances of the Willmore flow and using an invariant for triple-point-free spheres.
result Classification of initial surfaces with energy at most 12π leading to unavoidable singularities.
We investigate the initial value problem for the Einstein-Euler equations of general relativity under the assumption of Gowdy symmetry on T3, and we construct matter spacetimes with low regularity. These spacetimes admit, both, impulsive gravitational waves in the metric (for instance, Dirac mass curvature singularitie…
Re-initializing neural networks improves generalization but not as much as other techniques.
problem Understanding when and how re-initialization improves neural network performance.
method Empirical comparison of re-initialization with standard training and various regularization techniques.
result Re-initialization is beneficial for generalization but not as much as other techniques, especially when combined with careful tuning of hyperparameters.
Improved performance of factorized neural layers through spectral initialization and Frobenius decay.
problem Improving the performance of factorized neural layers in various deep learning contexts.
method Spectral initialization and Frobenius decay for initialization and regularization.
result Spectral initialization and Frobenius decay lead to improved performance across multiple deep learning settings.
Gradient flow with infinitesimal initialization converges to Greedy Low-Rank Learning for matrix factorization.
problem Understanding implicit regularization in gradient descent for matrix factorization.
method Theoretical and empirical analysis of gradient flow with infinitesimal initialization and Greedy Low-Rank Learning.
result Gradient flow with infinitesimal initialization is mathematically equivalent to Greedy Low-Rank Learning for depth-2 matrix factorization under reasonable assumptions.
Curve shortening flow's regularity depends on initial conditions after a certain time.
problem Understanding the regularity of evolving curves under curve shortening flow.
method Proposing and proving principles of controllable regularity based on initial conditions.
result No regularity estimate holds before a specific time, .
We reparametrize ReLU NNs as splines to understand their learning dynamics.
problem Understanding the learning dynamics and inductive bias of neural networks.
method Reparametrize ReLU NNs as continuous piecewise linear splines to study learning dynamics.
result Standard weight initializations yield very flat functions, leading to strength and type of implicit regularization.
Gradient descent recovers principal components of overparametrized asymmetric matrices without explicit regularization.
problem Asymmetric matrix factorization under overparametrization with minimal rank assumptions.
method Vanilla gradient descent with small random initialization and proper early stopping.
result Gradient descent produces the best low-rank approximation without explicit regularization.
We prove a conjecture of Tom Ilmanen's and Hubert Bray's regarding the existence of the outermost generalized apparent horizon in an initial data set and that it is outer area minimizing.
GCN and GPCA are mathematically connected, leading to improved node classification performance.
problem Improving node classification performance in semi-supervised settings.
method Established a mathematical connection between GCN and GPCA, demonstrating their equivalence and using this to design an effective initialization strategy.
result GPCA paired with a simple MLP achieves similar or better performance than GCN on semi-supervised node classification tasks.
The paper analyzes how gradient descent implicitly regularizes solutions in overparameterized neural networks, revealing depth-dependent regularization effects.
problem Understanding implicit regularization in overparameterized linear neural networks for regression problems.
method Analyzing the approximation error between gradient flow limit points and -minimization solutions, deriving tight upper and lower bounds.
result The approximation error decreases linearly for and at a slower rate for , linked to null space property constants.
Gradient descent in tensor factorization favors low-rank solutions.
problem Tackling implicit regularization in tensor factorization problems.
method Gradient descent with small random initialization for overparametrized tensor factorization.
result Gradient descent leads to implicit regularization towards low tubal rank solutions.
Proves higher regularity for anisotropic inverse mean curvature flow.
problem Higher regularity of solutions to anisotropic inverse mean curvature flow.
method Proves Harnack estimate and constructs smooth solutions from initial sets.
result Smooth solutions become smooth outside a compact set.
Convergence Analysis and Implicit Regularization of Feedback Alignment for Deep Linear Networkscs.LG
FA algorithm provides convergence guarantees for deep linear networks.
problem Training efficiency and convergence of deep neural networks.
method Theoretical analysis of Feedback Alignment (FA) algorithm for deep linear networks.
result Certain initializations lead to implicit anti-regularization, affecting learning effectiveness.
Softmax policy gradient methods converge at rate with constants depending on problem and initialization.
problem Understanding convergence rates of softmax policy gradient methods in tabular settings.
method Analysis of softmax policy gradient and entropy regularized policy gradient methods, using Łojasiewicz inequality and lower bounds.
result Entropy regularization improves convergence rate from to .
-means: Fighting against Degeneracy in Sequential Monte Carlo with an Application to Trackingstat.ML
For regular particle filter algorithm or Sequential Monte Carlo (SMC) methods, the initial weights are traditionally dependent on the proposed distribution, the posterior distribution at the current timestamp in the sampled sequence, and the target is the posterior distribution of the previous timestamp. This is techni…
Changing initialization scale affects deep model generalization, leading to memorization or improved performance.
problem Understanding how initialization scale impacts deep model generalization and memorization.
method Experimental setup with varying initialization scales, analysis of activation and loss functions, and development of an alignment measure.
result Increasing initialization scale leads to memorization, and decreasing it improves generalization, depending on activation and loss functions.
High regularity biharmonic wave maps shown to be locally well-posed.
problem Local wellposedness of biharmonic wave maps with high Sobolev regularity.
method Vanishing viscosity and parabolic regularization to prove existence; geometric nature exploited.
result Local wellposedness established in high Sobolev regularity.
The paper proves optimizability implies inequalities for sampling.
problem Optimizing functions via Gradient Flow and sampling from Gibbs measures.
method Gradient Flow and Lyapunov potentials to establish inequalities.
result Optimizability via Gradient Flow implies Poincaré and Log-Sobolev Inequalities.
Study spin-0 fields on n-dimensional Minkowski spacetimes, computing asymptotic charges.
problem Analyzing spin-0 fields on Minkowski spacetimes near infinity.
method Conformal geometry and Friedrich's cylinder at spatial infinity.
result Found infinitely many well-defined asymptotic charges in even dimensions, no charges in odd dimensions.
The paper defines function spaces on manifolds with bounded or singular geometries.
problem Defining function spaces on manifolds with various geometries.
method Introduces and analyzes Sobolev, Besov, and Bessel potential spaces on uniformly regular and singular Riemannian manifolds.
result Demonstrates maximal regularity for a linear parabolic problem on singular manifolds.
We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White's local regularity theorem for mean curvature …
Gradient Descent with small random initialization solves rank-1 matrix completion efficiently.
problem Matrix completion for rank-1 symmetric matrices.
method Gradient Descent with small random initialization.
result Gradient Descent converges to the ground truth for rank-1 symmetric matrix completion.
Consider a mean curvature flow of hypersurfaces in Euclidean space, that is initially graphical inside a cylinder. There exists a period of time during which the flow is graphical inside the cylinder of half the radius. Here we prove a lower bound on this period depending on the Lipschitz-constant of the initial graphi…
Study on the smoothness of solutions to a specific type of stochastic differential equation.
problem Regularity of solutions to mean-field -SDEs.
method Analysis of first and second order Fréchet differentiability in the random initial condition.
result Established the Fréchet differentiability of the solution and specified the corresponding equations.
Regularizes trajectory optimization with denoising autoencoders.
problem Trajectory optimization models are prone to inaccuracies.
method Uses a denoising autoencoder trained on the same trajectories.
result Improves planning with gradient-based and gradient-free optimizers.
We consider the unnormalized Yamabe flow on manifolds with conical singularities. Under certain geometric assumption on the initial cross-section we show well posedness of the short time solution in the -setting. Moreover, we give a picture of the deformation of the conical tips under the flow by providing an asym…
Why does training deep neural networks using stochastic gradient descent (SGD) result in a generalization error that does not worsen with the number of parameters in the network? To answer this question, we advocate a notion of effective model capacity that is dependent on {\em a given random initialization of the netw…
A regular F-manifold is an F-manifold (with Euler field) (M, \circ, e, E), such that the endomorphism {\mathcal U}(X) := E \circ X of TM is regular at any p\in M. We prove that the germ ((M,p), \circ, e, E) is uniquely determined (up to isomorphism) by the conjugacy class of {\mathcal U}_{p} : T_{p}M \rightarrow T_{p}M…
Proves existence and uniqueness of curvature motion for regular networks.
problem Existence and uniqueness of motion by curvature for regular networks.
method Proves existence and uniqueness using $W^{2- rac{2}{p}}_p$ initial data and investigates regularization effects.
result Proves existence and uniqueness of motion by curvature for regular networks.
Self-Expanding Solutions to the Mean Curvature Flow for Multiphase Surfaces with Regular Junctionsmath.DG
Study proves existence of expanding solutions for multiphase surfaces with regular junctions.
problem Existence of self-similar expanding solutions for multiphase surfaces with regular junctions.
method Proves existence of solutions for a multiphase surface with regular junctions using mean curvature flow.
result Multiple self-similar expanding solutions exist for the initial condition of a multiphase surface with regular junctions.
Gradient descent with early stopping achieves optimal sparse recovery.
problem Sparse regression with gradient descent and early stopping.
method Gradient descent on depth-N networks with early stopping.
result Implicit sparse regularization occurs with early stopping for general depth N.
Sharp results link DLN gradient flow to basis pursuit optimization and GHA phase transitions.
problem Understanding implicit regularization in Diagonal Linear Networks.
method Sharp convergence bounds and characterization of minimizers.
result Gradient flow of DLNs with tiny initialization approximates minimizers of basis pursuit optimization problem.
The Willmore Flow of Graphs with Boundary Data: Low-Regularity Initial Data and Global Convergencemath.AP
Analyzes Willmore flow for graphs with boundary data, proving existence and convergence.
problem Willmore flow of graphs with boundary conditions over bounded domains.
method Developed low-regularity theory, reformulated graphical equation, used time-weighted parabolic Hölder spaces.
result Proved short-time and global existence for initial data in and Lipschitz, with exponential convergence.
The paper introduces a multilevel initialization method for deep neural networks.
problem Training very deep neural networks with layer-parallel methods.
method Continuous interpretation of training as optimal control, using time-dependent ODEs for neural network discretization, and a refinement strategy across the time domain.
result The method creates deep networks with good initializations from coarser networks, reducing training time and providing regularization.
Large initial learning rate helps neural nets generalize better.
problem Understanding why large initial learning rates lead to better neural net generalization.
method Developed a proof for a two-layer network and demonstrated with experiments on CIFAR-10.
result Proved that a two-layer network trained with a large initial learning rate and annealing generalizes better than one trained with a small learning rate.
New method reduces variance in reinforcement learning.
problem High variance in reinforcement learning models.
method Functional regularization of deep policies to stabilize learning.
result Significantly reduced variance and improved stability.
Provable Emergence of Deep Neural Collapse and Low-Rank Bias in -Regularized Nonlinear Networkscs.LG
The paper connects neural collapse and low-rank bias in networks with L2 regularization.
problem Understanding the emergence of low-rank bias and neural collapse in L2-regularized networks.
method Unified theoretical framework linking TCV and rank of weight matrices, proving global optimality of DNC1, and establishing a benign landscape property.
result Zero TCV across intermediate layers minimizes representation cost under natural architectural constraints, and DNC1 is globally optimal.
We study a policy gradient method with L2 regularization for MAB problems.
problem Improving policy gradient methods for MAB problems with regularization.
method Investigate convergence of a policy gradient algorithm with L2 regularization for MAB.
result Prove convergence under appropriate technical hypotheses and show practical improvements.
In this paper we apply known techniques from semigroup theory to the Schrödinger problem with initial conditions. To this end, we define the regularized Schrödinger semigroup acting on a space-time domain and show that it is strongly continuous and contractive in with These results can easily be e…
Paper studies Einstein vacuum equations with low regularity data.
problem Einstein vacuum equations with low regularity initial data.
method Combines Klainerman-Szeftel-Rodnianski curvature theorem, Czimek's extension procedure, and global elliptic estimates.
result Time of existence controlled by low regularity bounds on curvature in .
On the set of points of smoothness for the value function for affine optimal control problemsmath.OC
We study the regularity properties of the value function associated with an affine optimal control problem with quadratic cost plus a potential, for a fixed final time and initial point. Without assuming any condition on singular minimizers, we prove that the value function is continuous on an open and dense subset of …
The paper improves the regularity and existence of pseudo Calabi flow.
problem Improving the smoothness and existence of pseudo Calabi flow.
method Analyzing the initial conditions and using volume form closeness to smooth metrics.
result The pseudo Calabi flow becomes smooth immediately and exists for all time under certain conditions.