The paper studies the consistency of mean curvature flow via volumetric varifolds.
problem Consistency of mean curvature flow.
method Discretization using volumetric varifolds and derivation of Brakke approximate equality.
result Derivation of a Brakke approximate equality involving varifold masses and approximate mean curvatures.
Constructs approximate mean curvature flows for general varifolds.
problem Mean curvature flow for general initial data.
method Approximation of mean curvature flows using varifolds and iterated push-forwards.
result Approximate mean curvature flow converges to a spacetime Brakke flow under certain conditions.
Approximates nonlocal curvature of curves using splines.
problem Approximating nonlocal curvature of planar curves.
method Extending nonlocal curvature definition, using incomplete beta function, and linear interpolating spline approximation.
result Nonlocal curvature of a planar curve can be approximated by a spline.
The study provides conditions for approximating Riemannian manifolds with polyhedral metrics.
problem Approximating Riemannian manifolds with polyhedral metrics.
method Conditions on curvature tensors for Lipschitz and local polyhedral approximations.
result Conditions are sufficient for local polyhedral approximations, conjectured to be sufficient for global approximations.
Algorithm approximates functions into manifolds with curvature bounds.
problem Approximating functions into manifolds with lower curvature bounds.
method Algorithm using manifold exponential and logarithm, with error bounds based on sectional curvature.
result Error bounds for nonnegative sectional curvature are similar to linear space approximations.
For an embedded submanifold Σ⊂RN, Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds Σ in the same way. For this purpose…
New game approximates mean curvature flow evolution.
problem Approximating geometric mean curvature flow evolution.
method Two-player zero-sum game with probabilistic elements.
result Value function approximates mean curvature flow.
Study approximates Riemannian manifolds using polyhedra.
problem Understanding Tullio Regge's approximation theorem.
method Proof of Regge theorem using polyhedra approximation.
result Integral of scalar curvature approximated by polyhedral curvature.
New method approximates curvature from symmetries in deep networks.
problem Hard to approximate curvature in large deep networks.
method Analytically averaging over group actions that leave the loss invariant to construct structured Hessian approximations.
result Structured Hessian approximations from single gradients can be estimated, stored, and inverted.
Smooth flows with surgery approximate weak mean curvature flows with spherical and neck-pinch singularities.
problem Approximating weak mean curvature flows with singularities using smooth flows.
method Combining Choi-Haslhofer-Hershkovits and Choi-Haslhofer-Hershkovits-White work on canonical neighbourhoods and barriers to flows with surgery.
result Smooth flows with surgery can approximate weak mean curvature flows with spherical and neck-pinch singularities.
Smooth curves with specific curvature can be closely approximated.
problem Approximating smooth curves with prescribed curvature.
method Application of h-principle to C1-dense approximation of curves. result Existence of C∞ knots with prescribed curvature. Smooth approximations bound dihedral angles of convex polytopes.
problem Bounding dihedral angles of convex polytopes.
method Approximating polytopes with smooth hypersurfaces and using geometric relations.
result Established lower bounds on dihedral angles.
Piecewise flat approximations for curvature in Euclidean and non-Euclidean spaces.
problem Approximating local extrinsic curvature on discrete manifolds.
method Constructing discrete curvature forms on piecewise flat manifolds, using weighted sums of hinge angles.
result Converges to smooth curvature values as mesh refinement occurs, favorably comparing with other discrete approaches.
Method approximates Lipschitz domains with smoother shapes.
problem Approximating bounded Lipschitz domains.
method Sequence of smooth, bounded domains with weak curvatures.
result Uniform isocapacitary estimates for approximating sets.
Smooth approximations near singularities of constant mean curvature surfaces are found.
problem Finding smooth approximations for constant mean curvature surfaces near singular points.
method Proving the existence of sequences of smooth CMC hypersurfaces converging to a given one in a ball centered at the singularity.
result Smooth approximations exist in a ball centered at the singularity of a CMC hypersurface.
In this paper, we study the second approximate Matsumoto metric on a manifold M. We prove that F is of scalar flag curvature and isotropic S-curvature if and only if it is isotropic Berwald metric with almost isotropic flag curvature.
We propose an efficient method for approximating natural gradient descent in neural networks which we call Kronecker-Factored Approximate Curvature (K-FAC). K-FAC is based on an efficiently invertible approximation of a neural network's Fisher information matrix which is neither diagonal nor low-rank, and in some cases…
New method approximates Gaussian curvature on discrete surfaces.
problem Approximating solutions to the prescribed Gaussian curvature problem.
method Discrete conformality and convex functional minimization.
result Efficient numerical method to compute solutions.
Study on curvature invariants near singularities of wavefronts.
problem Conditions for extendibility and boundedness of curvature invariants.
method Investigation of Gaussian curvature, Mean curvature, and principal curvatures near singularities.
result Relationship between convergence to infinity and uniform approximation of fronts.
Smoothly embed maps with controlled curvature errors.
problem Approximating short maps with smooth isometric immersions.
method δ-approximation of strictly short maps to C^∞-smooth isometric immersions with controlled curvature.
result Achieve isometric immersions with controlled curvature errors.
The paper improves convergence rates of curvature approximations using Regge elements.
problem Improving convergence rates of curvature approximations using Regge elements.
method Investigates the interplay between polynomial degree of curvature lifting and metric tensor degree in Regge finite element space.
result Higher convergence rates are achieved by reducing the polynomial degree of curvature lifting and using linear Regge elements.
We prove that every immersed C2-curve γ in Rn, n⩾3 with curvature kγ can be C1-approximated by immersed C2-curves having prescribed curvature k>kγ. The approximating curves satisfy a C1-dense h-principle. As an application we obtain the existence of C2-knots of arbitrary p…
We apply Tian's method in Kahler-Einstein problem to prove that a conic K\''ahler metric with lower Ricci curvature bound can be approximated by smooth K\''ahler metrics with the same lower Ricci curvature bound. Furthermore, conic singularities here can be along a simple normal crossing divisor.
Boosting Variational Inference improves posterior approximations with adaptive step-sizes.
problem Limited resources hinder the widespread adoption of Boosting Variational Inference.
method Characterized global curvature impact, introduced local curvature, and developed an approximate backtracking algorithm.
result New theoretical convergence rates and experimental validation demonstrate improved performance.
An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in R3 is obtained. This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite total curvature, that is to say, having finite total curvature on proper regions of fin…
Study 2D spaces with curvature, focusing on structure and approximations.
problem Characterize and understand 2D metric spaces with curvature constraints.
method Lipschitz homotopy approximations, curvature measures, convergence analysis.
result Established Gauss-Bonnet Theorem and characterized spaces.
The paper generalizes Riemann curvature for manifolds with discontinuous metrics.
problem Generalizing Riemann curvature for manifolds with discontinuous metrics.
method Proposes a generalized Riemann curvature tensor combining angle defects and jumps in second fundamental forms.
result The generalized curvature tensor approximates classical curvature for smooth approximations of metrics.
Smooths metrics with nonnegative scalar curvature near singular sets.
problem Approximating metrics with nonnegative scalar curvature near singularities.
method Ricci-DeTurck flow to approximate metrics.
result Approximated metrics converge to the original metric in C∞ away from the singular set. Second-order optimization methods such as natural gradient descent have the potential to speed up training of neural networks by correcting for the curvature of the loss function. Unfortunately, the exact natural gradient is impractical to compute for large models, and most approximations either require an expensive it…
Study on t-graphs with prescribed mean curvature in Heisenberg groups.
problem Existence and uniqueness of t-graphs with prescribed mean curvature. method Characterization of classical solutions without Dirichlet boundary data, conditions for uniqueness, approximation technique for non-constant mean curvature.
result Conditions for existence and uniqueness of t-graphs in Heisenberg groups. Develops a curvature-corrected tangent space method for manifold-valued data.
problem Generalizing real-valued data approximation to manifold-valued data.
method Systematic approach to developing global-geometry aware, computationally feasible approximation schemes.
result Proposes CC-tHOSVD for low-rank approximation of manifold-valued data.
We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with infinitesimal quads. It is shown that choosing the points of discrete nets on the smooth surface one can obtain second-order approximation g…
ViViT efficiently computes curvature for deep networks without approximations.
problem Efficiently computing curvature for deep networks without approximations.
method Leverages the GGN's low-rank structure without further approximations.
result ViViT allows for efficient computation of eigenvalues, eigenvectors, and directional derivatives.
We propose a modular extension of backpropagation for the computation of block-diagonal approximations to various curvature matrices of the training objective (in particular, the Hessian, generalized Gauss-Newton, and positive-curvature Hessian). The approach reduces the otherwise tedious manual derivation of these mat…
Ginger efficiently approximates curvature with linear complexity for neural networks.
problem Quadratic memory and cubic time complexity for computing curvature matrices in deep learning.
method Ginger uses eigendecomposition to maintain the inverse of the generalized Gauss-Newton matrix, achieving linear memory and time complexity.
result Ginger provides an effective and efficient curvature approximation for non-convex objectives.
Graph Laplacian approximates manifold eigenvalues with controlled curvature bounds.
problem Approximating eigenvalues of Laplace-Beltrami on manifolds with bounded Ricci curvature.
method Graph discretization of Riemannian manifolds with (ε,ρ)-approximation, proving eigenvalue convergence. result Graph Laplacian eigenvalues converge uniformly to manifold Laplacian eigenvalues as parameters approach zero.
A new method for optimizing deep neural networks using TKFAC.
problem Optimizing deep neural networks with second-order methods.
method Proposes Trace-restricted Kronecker-factored Approximate Curvature (TKFAC) for Fisher information matrix approximation.
result TKFAC improves performance on deep network architectures compared to state-of-the-art algorithms.
A discrete method approximates hyperbolic curvature flow in the plane.
problem Modeling wave phenomena in solid-liquid interfaces.
method Semidiscrete finite difference method for hyperbolic curvature flow.
result Error bounds for natural discrete norms are proven.
Duality principle for approximation of geometrical objects (also known as Eudoxus exhaustion method) was extended and perfected by Archimedes in his famous tractate "Measurement of circle". The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygon…
Better Hessian approximations improve influence function attributions in deep learning.
problem Influence functions are difficult to compute due to ill-conditioned Hessians, leading to poor data attribution performance.
method Investigated the impact of Hessian approximation quality on influence-function attributions in a controlled setting.
result Better Hessian approximations consistently yield better influence score quality.
Study on curvature equation in Heisenberg group with convex boundary.
problem Existence of solutions to prescribed mean curvature equation in sub-Finsler Heisenberg group.
method Finsler approximation scheme to prove existence of Lipschitz solutions.
result Existence of a Lipschitz solution for the Dirichlet problem.
Finite element method approximates scalar curvature in arbitrary dimensions.
problem Approximating scalar curvature using finite elements in arbitrary dimensions.
method Piecewise polynomial interpolants of a smooth Riemannian metric on a triangulated polyhedral domain.
result Finite element interpolants converge to scalar curvature with rate O(hr+1) in H−2(Ω) norm. Study proposes curvature flow model for Drosophila dorsal closure.
problem Modeling and understanding Drosophila dorsal closure during embryonic development.
method Curvature-based mathematical model, analysis of maximum-principle and integral-estimates, numerical approximation scheme.
result Established global existence and convergence for the model.
The paper solves a problem in metric geometry for disks with negative curvature.
problem Prescribing negative Gaussian curvature on the disk and boundary geodesic curvature.
method Variational approach and refined blow-up analysis for approximated problems.
result Existence of solutions under natural curvature assumptions.
The paper approximates Levi-Civita connection and curvature on 2D manifolds using finite elements.
problem Approximating Levi-Civita connection and curvature on 2D manifolds with finite elements.
method Using Regge finite elements, piecewise polynomial symmetric (0,2)-tensor fields, and distributional sense for non-regular tensors.
result Distributional quantities converge to their smooth counterparts under refinement of triangulation.
The paper studies how certain surfaces evolve over time.
problem Evolution of specific types of surfaces in high dimensions.
method Approximation by simpler problems to prove long-term existence.
result Long-time existence of the Hα-flow for specified surfaces. This paper improves HNNs by learning optimal curvature for better generalization.
problem Inappropriate curvatures in HNNs lead to suboptimal performance.
method Sharpness-aware curvature learning method to smooth loss landscape.
result Proposed method improves HNNs' generalization across various settings.
AQFC method estimates mesh curvatures using quadratic surfaces.
problem Estimating curvatures for irregular polygonal meshes.
method Local approximation of vertices and normals by quadratic surfaces, computed as implicit surfaces.
result AQFC provides robust curvature estimation for irregular meshes.