Noise causes learning plateaus in neural networks.
problem Plateau phenomena in online learning due to vanishing gradients.
method Analysis of stochastic gradient descent in multi-layer perceptrons.
result Noise induces synchronisation leading to strong plateaus.
This study investigates abrupt learning dynamics in Transformers, revealing plateau formation and internal representation collapse.
problem Abrupt learning in Transformers, particularly during the loss plateau.
method Investigates mechanisms of abrupt learning in shallow Transformers, focusing on attention maps and hidden states.
result Reveals plateau formation, internal representation collapse, and strong repetition bias in outputs.
Quantum machine learning faces 'laziness' and 'barren plateaus', but noise can mitigate the latter.
problem Quantum machine learning's loss function landscape issues.
method Theoretical analysis of quantum variational circuits, neural tangent kernels, and noise effects.
result Noise can mitigate barren plateaus in quantum machine learning.
Study approximates Plateau's laws using the Allen-Cahn equation.
problem Approximating Plateau's laws with the Allen-Cahn equation.
method Minimizing the Allen-Cahn energy under volume and spanning constraints.
result Energy minimizing solutions approximate Plateau-type singularities.
Gradient-free optimizers are ineffective on barren plateaus in quantum computing.
problem Effect of barren plateaus on gradient-free optimization in quantum computing.
method Numerical simulations and theoretical analysis of gradient-free optimization algorithms.
result Gradient-free optimizers are not effective in barren plateau landscapes due to exponentially suppressed cost function differences.
Paper proves stable minimal surfaces in 3D are flat.
problem Understanding stable minimal surfaces in 3D.
method Analyzes quadratic area growth and stability conditions.
result Stable minimal Plateau surfaces in 3D are flat.
Paper proves uniqueness of weak solutions for Plateau flow.
problem Proving uniqueness of weak solutions for Plateau flow.
method Used natural energy condition and alternative methods from Struwe.
result Proves uniqueness of weak solutions under natural condition.
Researchers solve a Plateau problem for maximal surfaces in pseudo-hyperbolic spaces.
problem Finding maximal surfaces with given boundary curves in pseudo-hyperbolic spaces.
method Defined and proved the existence of unique solutions using asymptotic Plateau problem and analysis of pseudo-holomorphic curves.
result Existence and uniqueness of maximal surfaces with specified boundary conditions.
Barren plateaus are not an average-case phenomenon, but a highly non-unique problem.
problem Avoiding barren plateaus in neural network training
method First-moment framework for initialization strategies
result Many families of inequivalent initialization strategies can avoid concentration
Solves Plateau problem for surfaces in pinched curvature manifolds.
problem Asymptotic Plateau problem for immersed surfaces in pinched curvature manifolds.
method Complete solution to asymptotic Plateau problem, providing dynamical stability of hypersurface laminations.
result Achieved complete solution to the asymptotic Plateau problem for immersed surfaces of constant extrinsic curvature in Cartan--Hadamard manifolds.
Solves area-minimizing surface problem for finite curves in H^2xR.
problem Asymptotic Plateau problem for area-minimizing surfaces.
method Complete solution for finite curves in $\BHH$.
result Fairly complete solution for finite curves in $\BHH$.
Unique solutions found for Plateau problems in smooth and continuous calibrations.
problem Finding unique solutions to the Plateau problem for specific types of currents.
method Boundary regularity theory for area-minimizing currents and unique continuation argument.
result Every compactly supported smoothly or continuously calibrated integral current is the unique solution to the Plateau problem for its boundary data.
Study on minimal submanifolds in curved spaces with unique solution to asymptotic Plateau problem.
problem Minimal submanifolds in negatively curved spaces with small curvature.
method Analysis of spheres at infinity and asymptotic Plateau problem.
result Complete minimal submanifolds bound a class of spheres with uniquely solvable asymptotic Plateau problem.
Derives equilibrium law for Plateau borders in wet soap films and foams.
problem Equilibrium law for Plateau borders in wet foams and films.
method Rigorous derivation using Gauss' capillarity theory, homotopic spanning condition, and effective compactness theorems.
result Sharp regularity properties of energy minimizers for Plateau borders in wet foams and films.
Spherical Plateau problem studies minimal surfaces in quotients of spheres.
problem Minimal surfaces in quotients of spheres.
method Metric currents, barycenter map method.
result Intrinsic uniqueness of solutions for negatively curved manifolds.
Solves Plateau's Problem in Heisenberg group for graphs.
problem Plateau's Problem in the Heisenberg group for intrinsic graphs.
method Geometric construction and calibration argument.
result Solves Plateau's Problem under smallness conditions.
Solves the asymptotic Plateau problem in hyperbolic space for specific curvature.
problem Existence of complete hypersurfaces with prescribed asymptotic boundary.
method Curvature estimates.
result Solves the problem for a wider range of curvature values.
Quantum models avoiding barren plateaus can also be efficiently simulated classically.
problem Understanding the limitations of barren plateaus in quantum computing.
method Analyzing commonly used models and their ability to be simulated classically.
result Many quantum models with barren plateau-free landscapes can also be efficiently simulated classically.
Plateau's problem is to show the existence of an area minimizing surface with a given boundary, a problem posed by Lagrange in 1760. Experiments conducted by Plateau showed that an area minimizing surface can be obtained in the form of a film of oil stretched on a wire frame, and the problem came to be called Plateau's…
Paper finds invariant solutions for Plateau problem in hyperbolic space.
problem Finding minimal surfaces with specific symmetries in hyperbolic space.
method Proved existence of foliations by invariant minimal surfaces, used to solve the Plateau problem.
result Existence of invariant minimal surfaces solving the asymptotic Plateau problem.
We give a solution of Plateau's problem for singular curves possibly having self-intersections. The proof is based on the solution of Plateau's problem for Jordan curves in very general metric spaces by Alexander Lytchak and Stefan Wenger and hence works also in a quite general setting. However the main result of this …
This is a survey of old and recent results about the asymptotic Plateau problem. Our aim is to give a fairly complete picture of the field, and present the current situation.
This research solves Plateau's problem for CRPC surfaces.
problem Constructing surfaces with constant ratio of principal curvatures.
method Proposed a family of surfaces containing a given minimal surface without flat points.
result Obtained a partial solution to Plateau's problem for CRPC surfaces.
New energy model avoids self-intersections in curve optimization.
problem Avoiding self-intersections in curve optimization under elastic boundary energies.
method Introduced Möbius-Plateau energy to minimize curve variations.
result Screw-like solutions are plentiful, ribbon-like solutions have constraints.
The plateau phenomenon, wherein the loss value stops decreasing during the process of learning, has been reported by various researchers. The phenomenon is actively inspected in the 1990s and found to be due to the fundamental hierarchical structure of neural network models. Then the phenomenon has been thought as inev…
Study on the Euler-Plateau energy with elastic modulus, focusing on minimizers and critical surfaces.
problem Minimizing the Euler-Plateau energy with elastic modulus.
method Analyzing the energy functional and its minimizers, considering different boundary conditions and topological constraints.
result Potential minimizers are highly dependent on physical rigidity parameters, and the area of critical surfaces can be computed from boundary data.
We present a novel and comprehensive approach to the study of the parametric Plateau problem for locally strictly convex (LSC) hypersurfaces of prescribed curvature for general convex curvature functions inside general Riemannian manifolds. We prove existence of solutions to the Plateau problem with outer barrier for L…
Generalizes embeddedness result for extreme curves.
problem Embeddedness of solutions to the Plateau problem for extreme curves.
method Generalization of Meeks-Yau's result.
result Generalization of embeddedness result.
Unified rigidity theorem for Plateau surfaces in Bn.
problem Rigidity of free-boundary minimal surfaces in Bn. method Analyzing conformal free-boundary minimal immersions of Plateau model cones.
result Every conformal free-boundary minimal immersion of the flat T-cone into Bn is congruent to the flat T-cone. It is extended a result due to B. Guan and J. Spruck on the asymptotic Plateau's problem for CMC radial graphs in hyperbolic space to horizontal CMC graphs.
Because of the relevance of the results, this paper is merged into the paper titled "On the Number of Solutions to Asymptotic Plateau Problem" (arXiv:math.DG/0505593) as a new section.
Following on from ``Hyperbolic Plateau problems'' (by the same author), we provide a complete geometric description of solutions to the Plateau problem (S,φ) when S is a compact Riemann surface with a finite number of points removed.
Smoothness of collapsed regions in soap films is proven, indicating wetted singularities.
problem Smoothness of collapsed regions in soap films.
method Study of generalized minimizers in capillarity model.
result Collapsed regions are smooth outside of dimensionally small singular sets.
Paper bounds surface diameter and solves Plateau-Douglas problem.
problem Bounding the diameter of compact surfaces and solving the Plateau-Douglas problem.
method Geometric argument based on Topping's diameter bound for closed surfaces.
result Explicit nonexistence criterion for the Plateau-Douglas problem.
Unique minimal surfaces near quadratic cones are identified.
problem Identifying minimal surfaces near quadratic cones.
method Analyzing minimal hypersurfaces inside the unit ball with perturbed boundary conditions.
result Minimal surfaces are uniquely determined by their boundary conditions.
Training task diversity improves ICL with linear attention.
problem Understanding the impact of training task diversity on in-context learning.
method Modeling training task vectors as a mixture of low-rank Gaussians.
result Our model explains why training with task diversity shortens the ICL plateau and achieves out-of-distribution generalization.
Paper estimates curvature of semi-convex hypersurfaces in hyperbolic space.
problem Estimating curvature of semi-convex hypersurfaces in hyperbolic space.
method Established C2 estimates using a new concavity inequality for hessian equations. result Derived C2 estimates for semi-convex complete hypersurfaces with constant σk curvature. We consider a complex Plateau problem for strongly pseudoconvex contours in non Kähler manifolds. A positive solution in the case of manifolds carrying a pluriclosed Hermitian metric forms is given. For the general case we propose a conjecture.
We apply Garnier's method to solve the Plateau problem for maximal surfaces in Minkowski 3-space. Our study relies on the improved version we gave of R. Garnier's resolution of the Plateau problem for polygonal boundary curves in Euclidean 3-space. Since in Minkowski space the method does not allow us to avoid the exis…
QCNNs avoid barren plateaus, making them trainable.
problem Exponentially vanishing gradients in QNNs.
method Graph-based method to analyze Haar-distributed unitaries.
result QCNNs do not exhibit barren plateaus, implying trainability.
In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions.
We describe a novel technique for solving the Plateau problem for constant curvature hypersurfaces based on recent work of Harvey and Lawson. This is illustrated by an existence theorem for hypersurfaces of constant Gaussian curvature in Rn+1.
The paper extends kernel ridge regression to product kernels and reveals new convergence behaviors.
problem Understanding kernel ridge regression in large dimensions with various kernels.
method Established a broad family of large dimensional kernels and derived convergence rates.
result Revealed new phenomena including minimax optimality, saturation effect, and multiple descent behavior.
Sharp theory of neural network scaling laws for hierarchical targets.
problem Learning hierarchical multi-index models in neural networks.
method Sharp information-theoretic scaling laws derived for two-layer neural networks.
result Optimal rates achieved by a simple spectral estimator.
New methods optimize training VQAs without barren plateaus, improving efficiency and applicability.
problem Barren plateaus in training variational quantum algorithms.
method Derive adaptive learning rates and use Gaussian kernels to optimize movement in parameter space.
result Optimized training methods outperform other routines and can train VQAs free of barren plateaus.
The paper solves a partial Plateau problem using H-mass.
problem Finding a surface of least area with a partially specified boundary.
method Minimizing H-mass over scans with boundary. result Existence of a rectifiable minimizer for the H-mass problem. Smooth minimizing hypersurfaces in 11D are generic, with singularities in higher dimensions.
problem Finding smooth minimizing hypersurfaces in high dimensions.
method Analyzing the Plateau problem and area minimization in integral homology.
result Smooth minimizing hypersurfaces are generic in 11D, with singularities in higher dimensions.
New bound shows variational algorithms may struggle with barren plateaus.
problem Barren plateaus in quantum loss landscapes.
method General bound on loss variance and gradient decay.
result Exponential decay of gradients in subregions of barren plateaus.