Lower bound found for flat norm minimizers' reach.
problem Finding the minimum flat norm for boundary shapes.
method Quantitative lower bound calculation.
result Established a lower bound on reach.
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A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
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48 results for “flat norm minimizers”
A theorem simplifies mass-minimizing flat chains' regularity.
problem Understanding the regularity of mass-minimizing flat chains.
method Simple condition for fundamental regularity principle.
result Fundamental regularity principle holds for mass-minimizing chains.
In this paper, we show that the complete scalar-flat Kahler metrics constructed by Abreu and the author on strictly unbounded toric 4-dimensional orbifolds have finite norm of the full Riemannian tensor. In particular, this answers a question of Donaldon's on the corresponding Generalized Taub-NUT metric on …
AMP regularization improves deep learning models by favoring flat minima.
problem Improving deep learning model generalization and avoiding overfitting.
method AMP regularization uses adversarial model perturbation to minimize a norm-bounded perturbation of the empirical risk.
result AMP regularization leads to state-of-the-art performance across various deep architectures.
Paper finds exact Hessian sharpness in deep matrix factorization.
problem Understanding the geometry of loss landscapes in deep matrix factorization.
method Presented the first exact expression for Hessian maximum eigenvalue.
result Spectral-norm balance is a sufficient condition for flatness in deep matrix factorization.
Flat minimal tori counterexamples refute Lu's second-gap conjecture.
problem Lu's second-gap conjecture about minimal surfaces in higher codimensions.
method Constructing closed embedded counterexamples for minimal surfaces.
result Constant values of realized by flat minimal tori are dense in , refuting the conjecture.
We study the multiscale simplicial flat norm (MSFN) problem, which computes flat norm at various scales of sets defined as oriented subcomplexes of finite simplicial complexes in arbitrary dimensions. We show that the multiscale simplicial flat norm is NP-complete when homology is defined over integers. We cast the mul…
The study sets limits on Weyl tensor norms for hypersurfaces and determines their homology.
problem Understanding the geometry and topology of hypersurfaces.
method Analyzes the Weyl tensor and Betti numbers for Riemannian manifolds.
result Determines the homology of almost conformally flat hypersurfaces.
The paper proves a rigidity theorem for minimal submanifolds in spheres with flat normal bundle.
problem Proving rigidity for minimal submanifolds in spheres with flat normal bundle.
method Explicit second-gap rigidity theorem for the squared norm of the second fundamental form.
result The theorem provides evidence for Chern's conjecture in higher codimension.
Rigidity theorem for ideal surfaces with flat boundary conditions.
problem Characterizing surfaces with flat boundary conditions.
method Analyzing a sixth order nonlinear elliptic PDE and flat boundary conditions.
result Surfaces with small second fundamental form and flat boundary conditions are planar.
We show that the recently introduced L1TV functional can be used to explicitly compute the flat norm for co-dimension one boundaries. While this observation alone is very useful, other important implications for image analysis and shape statistics include a method for denoising sets which are not boundaries or which ha…
Paper proves uniqueness of minimal hypersurfaces in specific domains.
problem Proving uniqueness of minimal hypersurfaces in constrained domains.
method Analyzing flat and compact free boundary minimal hypersurfaces in Euclidean balls and annular domains.
result Uniqueness of minimal hypersurfaces in unit Euclidean ball and annular domains.
Muon replaces matrix gradient with polar factor, optimizing flat spectrum updates
problem Optimization bias in matrix updates
method Using polar factor of gradient
result Muon update maximizes entropy among bounded updates
The paper explores how neural networks avoid overfitting by learning from flat minima.
problem How to prevent overfitting in neural networks by learning from flat minima.
method Study of one- and two-layer neural network models, derivation of algorithms focusing on wide flat regions.
result Wide flat minima coexist with narrower minima and critical points, and are associated with good minimizers.
New algorithms improve neural network generalization by finding flat minima.
problem Finding better generalization in neural networks through flat minima.
method Developed Entropy-SGD and Replicated-SGD algorithms to maximize flatness in the loss function.
result Consistently improved generalization error for various deep learning architectures.
A new method compares synthetic power networks to actual ones using multiscale flat norm.
problem Comparing synthetic power networks to actual ones due to lack of correspondence.
method Proposes a multiscale flat norm approach to compute distance between networks.
result The flat norm distance captures variations more accurately than Hausdorff distance.
Study on minimal hypersurfaces in a special normed space.
problem Characterizing minimal hypersurfaces in a specific normed space.
method Investigate translation and separable minimal hypersurfaces.
result New insights into the properties of minimal hypersurfaces.
The study pinches rigidity theorems for minimal submanifolds in spheres.
problem Pinching rigidity theorems for minimal submanifolds in spheres.
method Analyzes the shape operators and eigenvalues of submanifolds to prove rigidity conditions.
result If certain conditions are met, the normal bundle of the submanifold is flat.
The paper defines minimal norm tensors for curvature and divergence tensors, explaining Weyl and Cotten tensors.
problem Understanding curvature tensors and their minimal norm.
method Analyzing minimal norm tensors for third and fourth covariant tensors, including Riemannian curvature and divergence.
result Weyl tensor and Cotten tensor are identified as minimal norm tensors of Riemannian curvature and divergence tensors, respectively.
Study on minimal surfaces in a 3D space with 2m-norm.
problem Characterizing minimal surfaces in a specific geometric space.
method Examining translation, homothetical, and separable minimal surfaces.
result New insights into minimal surfaces in a 3D space with 2m-norm.
We construct a canonical element, called the refined analytic torsion, of the determinant line of the cohomology of a closed oriented odd-dimensional manifold M with coefficients in a flat complex vector bundle E. We compute the Ray-Singer norm of the refined analytic torsion. In particular, if there exists a flat Herm…
Proves spectral gap bounds for Teichmüller geodesics on flat surfaces.
problem Quantify spectral gaps for Teichmüller geodesics.
method Bounding spectral gaps in terms of geometric quantities on flat surfaces.
result Quantitative non-uniform hyperbolicity of Teichmüller geodesic flow.
Effective quantum dynamics on a thin Möbius strip approximated by a flat model.
problem Quantum dynamics on a Möbius strip with zero width.
method Norm-resolvent convergence to an unconventional flat model with explicit spectrum.
result Spectral properties of the curved Möbius strip are well approximated by a flat model.
Quantifies closeness of special Lagrangians under Floer conditions.
problem Estimating closeness of special Lagrangians.
method Floer theoretic conditions leading to quantitative estimates.
result Strong-weak uniqueness theorem for special Lagrangians.
Study shows closed Bach-flat manifolds with positive scalar curvature are locally spherical.
problem Characterizing closed Bach-flat manifolds with positive scalar curvature.
method Applied a different method to show local sphericality compared to previous complete non-compact cases.
result Closed Bach-flat manifolds with positive scalar curvature are locally spherical.
New optimization method helps models generalize better after achieving near-perfect training performance.
problem Models can achieve near-perfect training performance but fail to generalize well to unseen examples.
method GROKtimizer combines rapid convergence to interpolation with post-interpolation norm minimization using Critically Damped Momentum.
result GROKtimizer provides a quadratic speedup over classical gradient descent, offering a natural solution for selecting low-norm interpolating solutions.
New algorithms solve large-scale rank minimization problems efficiently.
problem Large-scale rank minimization problems.
method Define and apply bi-trace and tri-trace norms to rank minimization problems; design efficient linearized alternating minimization algorithms.
result Proved algorithms converge to critical points; provide RSC and MC error bounds.
Stable minimal hypersurfaces in math.DG
Flat stable minimal hypersurfaces in 5D are always flat.
problem Characterizing stable minimal hypersurfaces in higher dimensions.
method Analyzing properties of stable minimal hypersurfaces in \(\mathbf{R}^5\).
result Complete, two-sided stable minimal hypersurfaces in \(\mathbf{R}^5\) are flat.
Stable minimal hypersurfaces in math.DG
Flat minimal hypersurfaces in 4D space are always flat.
problem Understanding stable minimal hypersurfaces in 4D space.
method Proving stability and completeness lead to flatness.
result Complete, stable minimal hypersurfaces in 4D are flat.
Paper refines null space conditions for nuclear norm minimization in low-rank matrix recovery.
problem Establishing conditions for successful nuclear norm minimization recovery of low-rank matrices.
method Developed new null space conditions for nuclear norm minimization, proving their necessity and sufficiency.
result Weak null space condition is sufficient but not necessary for nuclear norm minimization recovery, providing a new necessary and sufficient condition.
Infinite width ReLU networks can approximate functions with bounded Euclidean norm.
problem Functions that can be approximated by ReLU networks with bounded Euclidean norm.
method Analyzing the minimal network norm required to approximate a given function.
result The minimal network norm for representing a function \( f \) is \( \max(\int |f''(x)| dx, |f'(-\infty) + f'(+\infty)|) \).
Every point on an asymptotically flat 3D space has a minimal plane nearby.
problem Finding minimal surfaces in asymptotically flat 3D spaces.
method Proving the existence of minimal planes for every point in the manifold.
result Every point in an asymptotically flat 3D space has a complete properly embedded minimal plane.
Smooth approximation of integral cycles in manifolds.
problem Approximating integral cycles in Riemannian manifolds.
method Approximation of integral cycles by smooth submanifolds with controlled area and singularities.
result Integral cycles can be approximated by smooth submanifolds with controlled area and singularities.
The paper defines curvature at infinity for flat manifolds.
problem Defining curvature at the boundary of flat manifolds.
method Constructing coordinates at infinity for asymptotically flat ends.
result A Weyl tensor and renormalized volume defined at infinity.
Hexagonal norm double bubble problem solved with minimal configurations.
problem Finding the optimal shapes for minimizing perimeter in hexagonal geometry.
method Elementary proof and geometric exclusions to simplify minimizer search.
result Existence of minimizing sets for volume ratio parameter α in (0,1].
Two flat structures on minimal surfaces lead to a unique harmonic function.
problem Understanding flat structures on minimal surfaces.
method Illustration of two independent flat structures and their harmonic function.
result Uniqueness of Enneper's surface captured by a harmonic function.
We address some theoretical guarantees for Schatten- quasi-norm minimization () in recovering low-rank matrices from compressed linear measurements. Firstly, using null space properties of the measurement operator, we provide a sufficient condition for exact recovery of low-rank matrices. This condition…
Study bounds on harmonic forms in hyperbolic 3-manifolds using Thurston norm and minimal surfaces.
problem Bounding the -norm of harmonic forms in hyperbolic 3-manifolds.
method Using Thurston norm and interaction with minimal surfaces.
result Generalizes inequalities of Brock-Dunfield and studies sharpness in closed and cusped cases.
Currents represent generalized surfaces studied in geometric measure theory. They range from relatively tame integral currents representing oriented compact manifolds with boundary and integer multiplicities, to arbitrary elements of the dual space of differential forms. The flat norm provides a natural distance in the…
Study calculates stable norm of slit tori using Farey sequence.
problem Computing the stable norm of slit tori.
method Explicit computations using the Farey sequence and gluing slit tori.
result Estimates the asymptotic counting of simple homology classes.
Paper proposes a new robust LDA method using L1,2-norm ratio minimization.
problem Outliers sensitivity in traditional LDA methods.
method L1,2-norm ratio minimization, novel efficient algorithm.
result The proposed method is effective and converges fast.
Formula proves monotonicity for anisotropic minimal hypersurfaces.
problem Understanding anisotropic minimal hypersurfaces.
method Proved a monotonicity formula under a sign assumption on the Minkowski norm.
result Monotonicity formula for anisotropic minimal hypersurfaces.
In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lore…
New method for tensor recovery with fewer samples.
problem Recovering low-TT-rank tensors from few samples.
method Minimizing a weighted sum of nuclear norms of unfoldings.
result Significantly fewer samples required for recovery.
We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surf…
Establishes uniform Hörmander estimates for flat line bundles on Kähler manifolds.
problem Estimating -operators for flat line bundles.
method Uniform -estimates for -operators on Kähler manifolds.
result Recovers Ueda's lemma for compact Kähler manifolds and generalizes to Ricci-flat manifolds.
Convergence of the Ricci flow on asymptotically flat manifolds with integral curvature pinchingmath.DG
Ricci flow on flat manifolds converges to Euclidean space under curvature pinching.
problem Curvature pinching on asymptotically flat manifolds.
method Ricci flow on asymptotically flat manifolds with integral curvature pinching.
result Ricci flow converges to flat Euclidean space for sufficiently pinched curvature.
Bach-Flat Kaehler Surfacesmath.DG
The study classifies Bach-flat metrics on Kaehler surfaces.
problem Classifying Bach-flat metrics on Kaehler surfaces.
method Riemannian metrics critical for Weyl curvature.
result Detailed results for each case in the classification.