Subharmonicity of Dirichlet energy proven for Kähler manifolds.
problem Subharmonicity of Dirichlet energy in Kähler families.
method Polarized family of compact Kähler manifolds, pluriharmonic maps, nonpositive complexified sectional curvature.
result Dirichlet energy is subharmonic in the parameter space.
Geometric Occam's Razor shapes deep learning solutions.
problem Understanding the regularization in over-parameterized neural networks.
method Analyzing the geometric model complexity and Dirichlet energy in neural networks.
result Over-parameterized neural networks are implicitly regularized by geometric model complexity.
Study classifies ruled surfaces critical to Dirichlet energy.
problem Identifying ruled surfaces critical to Dirichlet energy.
method Explicit parametrization of ruled surfaces.
result Classification of ruled surfaces as critical points of Dirichlet energy.
We extend short-time existence and stability of the Dirichlet energy flow as proven in a previous paper by the authors to a broader class of energy functionals. Furthermore, we derive some monotonely decreasing quantities for the Dirichlet energy flow and investigate an equation of soliton type. In particular, we show …
TPBS models improve robustness to overfitting with localized Dirichlet energy regularization.
problem Global Dirichlet energy-based regularization fails for TPBS models due to perfect interpolation.
method Propose local Dirichlet energy regularization and two inference estimators.
result TPBS models outperform neural networks in overfitting regimes and maintain competitive performance otherwise.
Graph Laplacians adapt to different manifold dimensions, while Dirichlet energies converge to a tensorized Dirichlet energy.
problem Understanding machine learning methods for data with varying intrinsic dimensions.
method Γ-convergence of graph Dirichlet energies and spectral convergence of graph Laplacians on intersecting manifolds of varying dimensions.
result Normalized Dirichlet energy converges to a tensorized Dirichlet energy that adapts to all dimensions simultaneously.
In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multi-valued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most (n−2), where n is the dimension of its domain. Almgren used this result in an essential way to show t…
The paper classifies surfaces in Euclidean space that minimize the Dirichlet energy.
problem Classifying surfaces that minimize the Dirichlet energy.
method Analyzing surfaces defined by the equation φxx+φyy=2Λ, where Λ is a real constant. result Surfaces that minimize the Dirichlet energy are either surfaces of revolution or of the type z=f(x)+g(y). In the 1980's, Almgren developed a theory of multi-valued Dirichlet energy minimizing functions on n dimensional domains and used it, in an essential way, to bound the Hausdorff dimension of the singular sets of area minimizing rectifiable currents of dimension n and codimension ≥2. Recent work of the second …
The paper classifies contact 3-manifolds with critical metrics and connects entropy to optimization.
problem Classifying contact 3-manifolds with critical metrics and understanding their entropy.
method Critical metrics optimization and entropy analysis.
result Anosov contact metrics' optimization is linked to Reeb dynamics and entropy.
The study finds surfaces with constant anisotropic mean curvature foliated by circles in Euclidean space.
problem Existence and geometric description of surfaces with constant anisotropic mean curvature.
method Analyzes surfaces with constant anisotropic mean curvature of the Dirichlet energy, proving existence and classifying them.
result Existence and geometric description of surfaces foliated by circles with zero anisotropic mean curvature.
In this paper, we consider multi-valued graphs with a prescribed real analytic interface that minimize the Dirichlet energy. Such objects arise as a linearized model of area minimizing currents with real analytic boundaries and our main result is that their singular set is discrete in 2 dimensions. This confirms (and p…
Derive Dirichlet scalar curvature energy functional variation formula
problem Dirichlet scalar curvature energy functional
method First variation formula
result Introduce Dirichlet-Einstein metrics
Let $f : U\subset\Rm \to \calQ_Q(\ell_2)$ be of Sobolev class W1,p, 1<p<∞. If f almost minimizes its p Dirichlet energy then f is Hölder continuous. If p=2 and f is squeeze and squash stationary then f is in VMO.
Geometric inequality linking Dirichlet and bienergy for maps between Riemannian manifolds.
problem Relating Dirichlet and bienergy for maps between Riemannian manifolds.
method Established a geometric inequality relating the Dirichlet energy and bienergy of smooth maps between Riemannian manifolds.
result Proved that E2(f)≥RicminE1(f) under specified conditions. Proves existence of special 2-spheres in curved 3-spaces.
problem Existence of constant mean curvature 2-spheres in Riemannian 3-spheres.
method Develops a min-max scheme for a weighted Dirichlet energy functional, using bi-harmonic approximation, derivative estimates, and Morse index estimates.
result Proves existence for almost every mean curvature and all for positively curved 3-spheres.
Lipschitz mappings found between Riemann surfaces with specific properties.
problem Finding globally Lipschitz mappings between doubly connected Riemann surfaces.
method Using a result from Iwaniec, Kovalev, and Onninen, the minimizer of the energy functional is shown to be locally Lipschitz and globally Lipschitz.
result The minimizer of the energy functional is a globally Lipschitz mapping.
New method clusters directed and undirected graphs without losing directional information.
problem Clustering directed graphs due to asymmetry in edge connectivity.
method Generalized Dirichlet Energy (GDE) and generalized spectral clustering (GSC).
result GSC outperforms existing methods in clustering accuracy and robustness.
The paper studies harmonic graphs in the Heisenberg group and their properties.
problem No analogous theorem exists for H-minimal surfaces in the Heisenberg group. method Introduced intrinsic Dirichlet energy and studied its critical points (contact harmonic graphs).
result Calibration condition and construction of energy-minimizing graphs with various singularities.
Study connects curvature to graph theory and reveals differences.
problem Exploring differences between Quadratic Orthogonal Bisectional Curvature and Real Bisectional Curvature.
method Real (1,1)--forms and Weitzenböck curvature operator used to represent graph Dirichlet energy.
result Curvature differences illuminated between Quadratic Orthogonal Bisectional Curvature and Real Bisectional Curvature.
Graph neural networks over-smooth when layers increase, reducing discriminative power.
problem Over-smoothing in graph neural networks reduces model performance as the number of layers increases.
method Analyzed over-smoothing in general graph neural network architecture using Dirichlet energy.
result The Dirichlet energy of embeddings converges to zero, leading to loss of discriminative power.
Study finds lower bounds for energy on fibred manifolds using fiberwise symmetrization.
problem Finding lower bounds for energy functionals on fibred manifolds.
method Established a framework for fiberwise symmetrization to find lower bounds.
result Proved a comparison theorem for the first eigenvalue of the Laplacian on warped product manifolds.
A Dirichlet k-partition of a domain U⊆Rd is a collection of k pairwise disjoint open subsets such that the sum of their first Laplace-Dirichlet eigenvalues is minimal. A discrete version of Dirichlet partitions has been posed on graphs with applications in data analysis. Both versions admit va…
The paper analyzes Laplace learning for Gaussian measure data in infinite dimensions, proving convergence.
problem Analyzing Laplace learning for infinite-dimensional Gaussian measure data.
method Minimizes Dirichlet energy on a graph constructed from the full dataset.
result Proves pointwise convergence of the graph Dirichlet energy for Gaussian measure data.
The variational problem for the functional F=21∥φ∗ω∥L22 is considered, where φ:(M,g)→(N,ω) maps a Riemannian manifold to a symplectic manifold. This functional arises in theoretical physics as the strong coupling limit of the Faddeev-Hopf energy, and may be regarded as a symplectic analogue of the D…
Paper defines a new functional for spinors on Euclidean manifolds.
problem No specific problem stated; focuses on a new functional.
method Variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary.
result Ricci flow is the gradient flow of the new functional.
Study index bounds for harmonic maps sequences with bubbles.
problem Upper and lower bounds of index and nullity for harmonic maps.
method Study limiting behavior of eigenfunctions of linearized operator; diagonalize index form with bilinear form varying with sequence.
result Obtain index bounds and show convergence of eigenfunctions on weak limit, bubbles, and neck regions.
Characterizes magnetic unit vector fields on Lie groups.
problem Classifying magnetic unit vector fields on Lie groups.
method Characterization through critical points of Landau Hall and Dirichlet energy functionals.
result Classification of all magnetic left invariant unit vector fields on 3-dimensional Lie groups.
Gradient flow method solves isoperimetric inequality for maps.
problem Finding maps with optimal enclosed area.
method Sobolev gradient flow for area-normalised Dirichlet energy.
result Solutions converge to a circle as time goes to infinity.
Study on energy of maps from K3 surface to flat orbifold.
problem Energy of maps from K3 surface to flat orbifold.
method Investigate Dirichlet energy of smooth maps and introduce an invariant.
result Ratio of energy to invariant converges to 1 for Foscolo's collapsing families.
Given a compact Riemannian manifold (M, g) and two positive functions ρ and σ, we are interested in the eigenvalues of the Dirichlet energy functional weighted by σ, with respect to the L 2 inner product weighted by ρ. Under some regularity conditions on ρ and σ, these eigenvalues are those of the operator …
Method computes harmonic and conformal maps from point clouds.
problem Computing maps from irregular point cloud data.
method Meshless method using cubic lattice approximations.
result Harmonic and conformal maps computed accurately.
Given a data set and a subset of labels the problem of semi-supervised learning on point clouds is to extend the labels to the entire data set. In this paper we extend the labels by minimising the constrained discrete p-Dirichlet energy. Under suitable conditions the discrete problem can be connected, in the large da…
Paper proposes learnable topological features for efficient phylogenetic inference.
problem Finding appropriate topological structures for phylogenetic inference tasks requires significant design effort and domain expertise.
method Combines raw node features with graph neural networks to automatically adapt to different tasks.
result Demonstrates effectiveness and efficiency on simulated and real data phylogenetic inference tasks.
The paper improves energy decay estimates for Dir-stationary Q-valued functions and applies them to Liouville-type theorems and continuity.
problem Improving energy decay estimates for Dir-stationary Q-valued functions.
method Establishing improved decay estimates and applying them to derive Liouville-type theorems and continuity.
result Dir-stationary Q-valued functions exhibit the Lebesgue property and reside in a generalized Campanato-Morrey space.
Motivated by a geometric problem, we introduce a new non-convex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show is strictly decreas…
The paper calculates the second variation of energy functions for families of canonically polarized manifolds.
problem Computing the second variation of energy functions for families of canonically polarized manifolds.
method Analyzing the Dirichlet energy of maps between fibers and using harmonic maps.
result The energy function is plurisubharmonic under certain curvature conditions.
Maximizes capacity of extensions with fixed boundary data.
problem Maximizing the capacity of extensions with nonnegative scalar curvature.
method Using the method of Lagrange multipliers on the constraint space of scalar-flat extensions.
result Derives variational condition for maximal capacity extensions and proves they have constant scalar curvature.
Dirac-harmonic maps are critical points of a fermionic action functional, generalizing the Dirichlet energy for harmonic maps. We consider the case where the source manifold is a closed Riemann surface with the canonical Spin^c-structure determined by the complex structure and the target space is a Kaehler manifold. If…
Study infinite circle patterns in the Weil-Petersson class using discrete harmonic functions.
problem Characterize infinite circle patterns in the Weil-Petersson class.
method Investigate circle patterns parameterized by discrete harmonic functions of finite Dirichlet energy, equipped with a Riemannian metric.
result Induced quasiconformal homeomorphisms from the unit disk to itself belong to the Weil-Petersson class.
AIR-Net adapts low-rank regularization dynamically for better image completion.
problem Fixed low-rank regularization limits adaptability to different images.
method AIR-Net uses adaptive and implicit regularization parameterized by a dynamic Laplacian matrix.
result AIR-Net enhances implicit regularization and outperforms fixed methods in non-uniform missing data scenarios.
We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed homotopy class and all hyperbolic metrics on the surface. We give explicit example…
We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of the Lipschitz functions. …
For any n>1 we give an explicit example of an n-axially symmetric Cartesian current in B^3 x S^2 with non-trivial vertical part and non-constant graph part minimizing the relaxed Dirichlet energy among the n-axially symmetric Cartesian currents with the same boundary. This stands in sharp contrast with a results of Har…
New metrics reveal oversmoothing in GNNs more accurately than traditional methods.
problem Oversmoothing in graph neural networks reduces model performance.
method Rank-based metrics to measure oversmoothing in GNNs.
result Rank-based metrics consistently capture oversmoothing, while energy-based metrics often fail.
We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiati…
We introduce a novel harmonic analysis for functions defined on the vertices of a strongly connected directed graph of which the random walk operator is the cornerstone. As a first step, we consider the set of eigenvectors of the random walk operator as a non-orthogonal Fourier-type basis for functions over directed gr…
New measure shows various training techniques control model complexity.
problem Understanding how to control model complexity in deep learning.
method Developed geometric complexity measure and demonstrated its effectiveness.
result Many training techniques control geometric complexity, providing a unified framework.