We prove existence and regularity of minimizers for Hölder densities over general surfaces of arbitrary dimension and codimension in \(\R^n \), satisfying a cohomological boundary condition, providing a natural dual to Reifenberg's Plateau problem. We generalize and extend methods of Reifenberg, Besicovitch, and Adams,…
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48 results for “Reifenberg method”
Characterizes submanifolds in terms of Reifenberg flatness.
problem Characterizing submanifolds in .
method Reifenberg type characterization and integral quotient condition.
result Characterization equivalent to Reifenberg-flatness with approximating planes.
Introduces methods to approximate complex sets by simpler Euclidean spaces.
problem Approximating complex sets by Euclidean subspaces on all scales.
method Reifenberg theory and Rectifiable Reifenberg theorem.
result Measures can be decomposed into rectifiable and non-rectifiable parts.
Theorem generalizes Reifenberg's for measures with bounds on β-numbers.
problem Bounding measures away from k-rectifiable sets with β-numbers.
method Assumptions on Jones' β-numbers to measure closeness to subspaces.
result Effective measure bounds on μ away from a closed k-rectifiable set.
The article studies effective Reifenberg theorems in Hilbert and Banach spaces.
problem Understanding measures in Hilbert and Banach spaces.
method Analyzes Reifenberg theorems for measures in Hilbert and Banach spaces, providing conditions for rectifiability.
result Conditions for rectifiability of measures in Banach spaces, including a power gain for uniformly smooth spaces.
The paper extends regularity for -minimizing maps using a Reifenberg Theorem.
problem Quantitative regularity of -minimizing maps between Riemannian manifolds.
method Stratification of singular points based on almost-symmetries, followed by application of a Reifenberg-type Theorem.
result Upper bound on the Minkowski content of the singular set, and -rectifiability of the singular set.
Analyzes soap films using BV functions and covering spaces.
problem Solving Plateau's problem with soap films.
method Covering space method with constrained BV functions.
result Examples of soap films not modelable with Reifenberg method.
This paper improves Reifenberg's theorem for almost calibrated sets, ensuring rectifiability with volume bounds.
problem Improving the rectifiability of sets that are close to subspaces under certain calibrations.
method Using ε-calibrations and positivity conditions, the paper shows that almost calibrated sets are rectifiable with volume bounds.
result Almost calibrated sets are rectifiable with uniform volume bounds.
Proof of Reifenberg theorem in metric spaces, expanding on Cheeger and Colding's work.
problem Proving the Reifenberg theorem in metric spaces using Gromov-Hausdorff distance.
method Detailed proof of Cheeger and Colding's result, expanding on their arguments.
result BiLipschitz version of the Reifenberg theorem in metric spaces.
The study defines a canonical nilpotent structure for certain collapsed manifolds.
problem Understanding the structure of collapsed Riemannian manifolds.
method Analyzes the nilpotent structure of manifolds with bounded Ricci curvature and Reifenberg local covering geometry.
result A canonical nilpotent structure can be defined and uniquely determined over regular limit spaces.
Harmonic maps to Euclidean buildings have rectifiable singular strata.
problem Understanding the structure of singular points for harmonic maps.
method Defining singular strata and proving rectifiability using the rectifiable Reifenberg program.
result Rectifiability of singular strata for harmonic maps into -connected complexes.
In this paper we study the regularity of stationary and minimizing harmonic maps between Riemannian manifolds. If $S^k(f)\equiv\{x\in M: \text{ no tangent map at $x$ is }k+1\text{-symmetric}\}$ is -stratum of the singular set of , then it is well known that , howeve…
In 1960 Reifenberg proved the topological disc property. He showed that a subset of which is well approximated by -dimensional affine spaces at each point and at each (small) scale is locally a bi-Hölder image of the unit ball in . In this paper we prove that a subset of which is well approximated b…
The paper proves topological finiteness for surfaces with finite Willmore energy.
problem Understanding the topology of surfaces with finite Willmore energy.
method Combining Allard regularity theorem and Reifenberg's topological disk theorem.
result Topological finiteness for a class of properly immersed surfaces with finite Willmore energy.
We study the existence and uniqueness of smooth mean curvature flow, in arbitrary dimension and co-dimension, emanating from so called -dimensional Reifenberg flat sets in . Our results generalize the ones from a previous paper by the author, in which the co-dimension one case (i.e. $…
In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in starting from any -dimensional -Reifenberg flat set with sufficiently small. More precisely, we show that the level set flow in such a situation is non-fattening and …
We study here limit spaces , where the have a lower Ricci curvature bound and are volume noncollapsed. Such limits may be quite singular, however it is known that there is a subset of full measure $\cR(Y)\subseteq Y$, called {\it regular} points, along with c…
Paper provides estimates for varifolds with critical mean curvature.
problem Estimating tilt-excess on varifolds with critical mean curvature.
method Generalizing Lipschitz approximation and Sobolev-Poincaré estimates to almost-integral rectifiable varifolds.
result VMO-type estimates for quadratic tilt-excess on varifolds with critical mean curvature.
The paper proves stability of Ricci flow for certain initial conditions.
problem Stability of Ricci flow for non-smooth initial metrics.
method Analyzes stability of Ricci flows starting from Reifenberg spaces with bounded curvature.
result Smooth three-dimensional, uniformly Ricci-pinched manifolds are either compact or flat.
Unified proof of smooth fibration theorems for collapsed manifolds.
problem Smooth fibration theorems for collapsed manifolds with Ricci curvature bounded below.
method Generalized Reifenberg condition and transformation technique for almost splitting maps.
result Unified proof of smooth fibration theorems in many previous works.
The paper proves fibration theorems for manifolds with almost nonnegative Ricci curvature.
problem Proving fibration theorems for manifolds with specific curvature conditions.
method Using equivariant regularity theorems and Gromov-Hausdorff convergence.
result Closed manifolds with certain curvature conditions fiber over a -torus.
We adapt to an infinite dimensional ambient space E.R. Reifenberg's epiperimetric inequality and a quantitative version of D. Preiss' second moments computations to establish that the set of regular points of an almost mass minimizing rectifiable chain in is dense in its support, whenever the group of …
Study of collapsed manifolds with bounded Ricci curvature and non-collapsed universal cover.
problem Understanding collapsed manifolds with specific Ricci curvature properties.
method Ricci flow techniques applied to non-collapsed universal cover.
result Partial extension of nilpotent structural results to global Ricci bounded covering geometry.
Spanning via Cech Cohomologymath.DG
Plateau's problem is to find a surface with minimal area spanning a given boundary. In 1960, Reifenberg and Adams developed a definition for "span" using Čech homology, and variants of this definition have been used ever sense. However, limitations of Čech homology resulted in the lack of a natural definition for a bou…
The paper studies harmonic map flows and proves rectifiability of singular sets.
problem Understanding the structure of singular sets in harmonic map flows.
method Investigates the stratification theory for suitable solutions using tangent measures.
result Each time slice of the singular set is rectifiable.
This paper studies limits of aspherical manifolds with specific curvature conditions.
problem Understanding the Gromov-Hausdorff limits of aspherical manifolds with given curvature constraints.
method Analyzing sequences of compact manifolds with Ricci curvature or sectional curvature conditions, and using diffeomorphism or homeomorphism properties.
result If the manifolds are diffeomorphic or homeomorphic to nilmanifolds, their limits are also diffeomorphic or homeomorphic to nilmanifolds.
A canonical diffeomorphism is constructed for manifolds near spheres.
problem Constructing a canonical diffeomorphism for manifolds near spheres.
method Using the first -eigenfunctions of the manifold, a map is constructed and shown to be a diffeomorphism with a uniform bi-Hölder estimate.
result The constructed diffeomorphism is canonical and satisfies a uniform bi-Hölder estimate, which is sharp and cannot be improved to a bi-Lipschitz estimate.
Study Ricci flows on manifolds, proving they behave like self-similar solutions and confirming a conjecture.
problem Understanding the behavior of Ricci flows on higher-dimensional manifolds.
method Analyzing -dimensional Ricci flows with non-negative Ricci curvature, starting at metric cones.
result Ricci flows behave like self-similar solutions up to an exponential error in time.
(Log-)epiperimetric inequality and regularity over smooth cones for almost Area-Minimizing currentsmath.AP
New epiperimetric inequality for cones with improved regularity results.
problem Regularity of almost area-minimizing currents at singular points.
method Flowing in radial direction for cones with isolated singularities.
result New ε-regularity result for almost area-minimizing currents.
The paper proves a transformation theorem under a monotone property of almost Euclidean factors of geodesic balls.
problem The non-increasing property of numbers of almost Euclidean factors of geodesic balls.
method Proves a transformation theorem under a non-decreasing property compared to the non-increasing property.
result Shows that for a manifold with nonnegative Ricci curvature, if its universal cover is polar at infinity and the number of almost Euclidean factors is monotone, then its fundamental group is finitely generated and virtually abelian.
A mean curvature flow starting from a closed embedded hypersurface in must develop singularities. We show that if the flow has only generic singularities, then the space-time singular set is contained in finitely many compact embedded -dimensional Lipschitz submanifolds plus a set of dimension at most …
The paper proves topological stability between RCD spaces and Riemannian manifolds.
problem Proving topological stability between RCD spaces and Riemannian manifolds.
method Using Gromov-Hausdorff distance and regular homeomorphisms, the paper constructs a map between spaces.
result There exists a regular homeomorphism between RCD spaces and Riemannian manifolds under certain conditions.
New non-canonical flows found via parabolic Allen-Cahn equations.
problem Existence of non-canonical mean curvature flows inside fattening regions.
method Construction of non-canonical flows as limits of parabolic ε-Allen-Cahn solutions.
result First examples of non-outermost, non-canonical integral Brakke motions.
We prove a new kind of estimate that holds on any manifold with lower Ricci bounds. It relates the geometry of two small balls with the same radius, potentially far apart, but centered in the interior of a common minimizing geodesic. It reveals new, previously unknown, properties that all generalized spaces with a lowe…
If one considers an integral varifold with bounded mean curvature, and if $S^k(I)\equiv\{x\in M: \text{ no tangent cone at $x$ is }k+1\text{-symmetric}\}$ is the standard stratification of the singular set, then it is well known that . In complete generality nothing else is known about …
Improved iterative methods for risk parity portfolio weights.
problem Solving for portfolio weights in risk parity allocation.
method Enhanced CCD and Newton methods, including a rescaling step and improved initial guess.
result Improved CCD method is the best, three times faster with 40% fewer iterations.
We describe a novel optimization method for finite sums (such as empirical risk minimization problems) building on the recently introduced SAGA method. Our method achieves an accelerated convergence rate on strongly convex smooth problems. Our method has only one parameter (a step size), and is radically simpler than o…
A new method combines Laplace and Variational Bayes for scalable inference.
problem Complex models and large datasets make exact inference infeasible.
method Low-Rank Variational Bayes Correction (VBC) using Laplace method and Variational Bayes correction in a lower dimension.
result The method ensures scalability in both model complexity and data size.
Unified framework for model explanation methods based on feature removal.
problem Unclear relationships and preferences among various model explanation methods.
method Characterizes removal-based explanations along three dimensions.
result Unified 26 existing methods, including widely used approaches.
This work reviews and evaluates methods for predicting prediction intervals in regression problems.
problem Calibration of prediction intervals in regression problems.
method Four classes of methods: Bayesian, ensemble, direct interval estimation, and conformal prediction.
result Conformal prediction can be used as a general calibration procedure.
Derives kernel PCA with Nyström method for scalability.
problem Scalability of kernel PCA.
method Nyström method for kernel PCA.
result Provides scalable alternative to full kernel PCA.
In this paper, the author considers the numerical computation of CVA for large systems by Mote Carlo methods. He introduces two types of stochastic mesh methods for the computations of CVA. In the first method, stochastic mesh method is used to obtain the future value of the derivative contracts. In the second method, …
Develops a fast method for pricing American options under variance gamma model.
problem Inefficient methods for pricing American options under variance gamma model.
method Inspired by quadratic approximation method, uses machine learning on pre-calculated quantities to reduce error.
result Proposed method is efficient and accurate for practical use.
Pricing Derivatives under Multiple Stochastic Factors by Localized Radial Basis Function Methodsq-fin.CP
Two RBF methods solve complex financial derivatives pricing problems.
problem Pricing derivatives in models with multiple stochastic factors.
method Radial Basis Function Partition of Unity and Radial Basis Function generated Finite Differences methods.
result Both methods achieve high accuracy and are efficient for solving multi-dimensional PDEs.
New method combines spectral and sparse methods for Gaussian processes.
problem Efficiently fitting Gaussian processes to large datasets.
method Orthogonally decoupled variational Fourier features.
result Competitive performance on synthetic and real-world data.
Simple stochastic Newton and cubic Newton methods with fast convergence.
problem Minimizing large numbers of smooth and strongly convex functions.
method Stochastic Newton and cubic Newton methods with simple local linear-quadratic rates.
result Local linear-quadratic convergence results with fast adaptation to problem's curvature.
Improved spectral methods of moments for robust latent variable model learning.
problem Limited robustness of spectral methods of moments to model misspecification.
method Hierarchical approach using approximate joint diagonalization instead of tensor decomposition.
result Our method outperforms previous tensor decomposition methods in speed and model quality.
VAN method optimizes learning tasks with unified methods.
problem Optimizing learning tasks in active and reinforcement learning.
method Variational Adaptive-Newton method that unifies optimization, inference, and evolution strategies.
result VAN performs well on various learning tasks.