The paper improves SVM learning rates for anisotropic Gaussian kernels.
problem Nonparametric regression with anisotropic Gaussian kernels.
method Establishing almost optimal learning rates for functions in anisotropic Besov spaces.
result Optimal learning rates up to logarithmic factors, faster than Sobolev space-based rates.
New method models dewetting of anisotropic particles using numerical techniques.
problem Modeling dewetting dynamics of particles with varying surface energies.
method Level set numerical approach with convolution kernels to handle anisotropic interfacial energies.
result Validated numerical scheme supports merging and splitting of interfaces.
Anisotropic neural network selects relevant features from datasets.
problem Reduction of irrelevant features improves model interpretability and performance.
method General Regression Neural Network with an anisotropic Gaussian Kernel.
result The method robustly selects features from simulated and real-world datasets.
New MMD statistic detects differences between locally low-dimensional distributions.
problem Detecting differences between two locally low-dimensional distributions.
method Anisotropic kernel-based Maximum Mean Discrepancy (MMD) statistic.
result Consistent test under mild assumptions, with finite-sample lower bound of testing power.
Proposes a new CNN for meshes that can handle orientation.
problem Isotropic kernels in graph convolutions are insensitive to mesh geometry.
method Introduces gauge equivariant kernels and geometric message passing.
result Significantly improved expressivity over conventional GCNs.
Paper proves geodesics and focal points unchanged by conformal changes in pseudo-Finsler manifolds.
problem Understanding the behavior of lightlike geodesics in pseudo-Finsler manifolds.
method Used Chern connection, anisotropic calculus, and critical points of energy functional to prove invariance.
result Lightlike geodesics and focal points are preserved by anisotropic conformal changes.
A fast method learns plasma collision kernels from simulations, improving kinetic models.
problem Improving kinetic models for plasma dynamics beyond the weakly coupled regime.
method Data-driven collisional operator, fast spectral separation method.
result Accurately captures plasma dynamics in moderately coupled regime.
We provide bounds for kernel matrices and new approximations for high-dimensional data.
problem Approximating high-dimensional empirical kernel matrices.
method Decoupling results for U-statistics and non-commutative Khintchine inequality.
result New tighter approximations for inner-product kernel matrices.
New kernel interprets 3D anisotropic data with rotations and improved predictions.
problem Capturing rotated anisotropy in 3D spatial fields.
method Introduces a Lie-algebraic kernel with three principal length-scales and an explicit rotation.
result Posterior recovers rotated anisotropy and improves prediction over axis-aligned kernels.
Study on KRR with power-law data, showing better sample complexity.
problem High-dimensional kernel ridge regression with anisotropic power-law covariance.
method Explicit characterization of kernel spectrum and asymptotic analysis of excess risk.
result Sample complexity is governed by effective dimension, not ambient dimension.
Deep learning performs well on high-dimensional data with anisotropic smoothness.
problem Understanding the performance of deep learning on high-dimensional datasets with varying smoothness.
method Investigated approximation and estimation errors in anisotropic Besov spaces.
result Deep learning's performance depends on the average smoothness, avoiding curse of dimensionality.
A new kernel improves statistical surrogates for stochastic manifolds with diverse data.
problem Handling statistical surrogates for stochastic manifolds with heterogeneous data.
method A transient anisotropic kernel is introduced to improve statistical surrogates for stochastic manifolds with heterogeneous data.
result The transient anisotropic kernel provides a better representation of statistical dependencies in the learned probability measure.
New method for causal inference with observed covariates improves learning rates.
problem Causal inference with observed covariates in nonparametric instrumental variable regression.
method Introduces novel Fourier measure for partial smoothing and adapts kernel lengthscales for anisotropic smoothness.
result Upper and lower learning rates for KIV-O show interpolation between NPIV and NPR rates.
Study heat kernel coefficients in Bianchi IX gravity models.
problem Analyzing gravitational wave properties in Bianchi IX models.
method Algebraic geometry and motives applied to heat kernel coefficients.
result Coefficients are algebro-geometric periods of motives.
GPR enhances materials discovery by automating parameter space exploration.
problem Automating exploration of large, high-dimensional parameter spaces in materials science.
method Gaussian process regression with inhomogeneous measurement noise and anisotropic kernels.
result Importance and benefits of tuning GPR for materials science experiments.
In this paper, we investigate a holonomy invariant elliptic anisotropic surface energy for hypersurfaces in a complete Riemannian manifold, where "holonomy invariant" means that the elliptic parametric Lagrangian (i.e., a Finsler metric) of the Riemannian manifold used to define the anisotropic surface energy is consta…
Paper proves anisotropic Minkowski inequality and related inequalities.
problem Proving anisotropic Minkowski inequality and related inequalities.
method Utilizes a nonlinear potential theoretic approach.
result Sharp anisotropic Minkowski inequality and related inequalities proved.
New Kelvin transform for anisotropic elliptic problems.
problem Semilinear and quasilinear anisotropic elliptic problems.
method Introducing a new Kelvin-type transform in the anisotropic setting.
result New insights into anisotropic elliptic problems.
Paper solves anisotropic capillary Minkowski problem for p ≥ 1.
problem Anisotropic capillary convex bodies and their properties.
method Introduced anisotropic capillary p-sum and computed variations of quermassintegrals. result Solved the anisotropic capillary Lp-Minkowski problem for p≥1. Paper proves rigidity results for anisotropic capillary hypersurfaces.
problem Rigidity of anisotropic capillary hypersurfaces.
method New Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces.
result Uniqueness of solution to anisotropic Orlicz-Christoffel-Minkowski problem.
Paper solves Minkowski problem for anisotropic p-torsional rigidity.
problem Solving the Minkowski problem for anisotropic p-torsional rigidity.
method Using the anisotropic p-Laplacian equation, presenting sufficient and necessary conditions for existence. result Presented sufficient and necessary conditions for the existence of a solution.
Anisotropic min-max theory constructs stable minimal surfaces in 3-manifolds.
problem Constructing stable anisotropic minimal surfaces in 3-manifolds.
method Anisotropic min-max theory, removable singularity theorems.
result Constructs stable anisotropic minimal surfaces in 3-manifolds without singularities.
Unique ancient solutions found for anisotropic curve shortening flow.
problem Finding unique solutions for anisotropic curve shortening flow.
method Constructing translating and ancient solutions under given conditions.
result Unique ancient and translating solutions found for anisotropic curve shortening flow.
New method polarizes anisotropic Heisenberg groups.
problem Polarizing anisotropic Heisenberg groups.
method Implementing a technique to polarize anisotropic Heisenberg groups.
result New class of polarizable Carnot groups expanded.
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 paper, we study the anisotropic Minkowski problem. It is a problem of prescribing the anisotropic Gauss-Kronecker curvature for a closed strongly convex hypersurface in Euclidean space as a function on its anisotropic normals in relative or Minkowski geometry. We first formulate such problem to a Monge-Ampére t…
A new kernel-based nonconformity score improves multivariate prediction regions.
problem Tackling the challenge of compressing multivariate residual vectors into scalars while preserving geometric structure.
method Introducing a Multivariate Kernel Score (MKS) that decomposes into an anisotropic MMD, providing finite-sample coverage guarantees and convergence rates.
result The MKS produces prediction regions that explicitly adapt to geometric structure, reducing volume compared to ellipsoidal baselines.
Motivated by the study of wave fronts in anisotropic media, we propose an incidence geometry of anisotropic spheres in a Finsler-Minkowski space. An anisotropic version of the Laguerre functional is considered. In some circumstances, this functional can be used to determine that two wavefronts observed at distinct time…
We introduce the anisotropic tensor calculus, which is a way of handling with tensors that depend on the direction remaining always in the same class. This means that the derivative of an anisotropic tensor is a tensor of the same type. As an application, we show how to define derivations using anisotropic linear conne…
A general approach to formulation of supergravity in higher order anisotropic superspaces (containing as particular cases different supersymmetric extensions and prolongations of Riemann, Finsler, Lagrange and Kaluza--Klein spaces) is given. We analyze three models of locally anisotropic supergravity.
Paper explains how early stopping helps distillation in overparameterized neural networks.
problem Understanding how overparameterized neural networks can improve with early stopping.
method Introducing Anisotropic Information Retrieval (AIR) to justify early stopping in distillation.
result Self-distillation algorithm improves over just early stopping, leading to better generalization.
Develops a PDE approach to constructing nontrivial anisotropic surfaces.
problem Min-max construction of anisotropic surfaces.
method PDE-based approach to anisotropic surface energies.
result Construction of an anisotropic min-max hypersurface.
Paper studies stability of curved surfaces in a half-space.
problem Stability of anisotropic capillary hypersurfaces in a half-space.
method Analyzes weak stability and proves Bernstein-type theorems.
result Compact hypersurfaces are stable if and only if they are a truncated Wulff shape.
Characterizes paths minimizing anisotropic lengths in Euclidean space.
problem Finding paths of minimal anisotropic length between points.
method Characterization through geometric connection to anisotropic isoperimetric set.
result Established a connection between minimizing paths and anisotropic isoperimetric geometry.
Anisotropic minimal graphs over half-spaces are flat.
problem Characterizing minimal graphs over half-spaces.
method Maximum principle and fully nonlinear PDE theory.
result Anisotropic minimal graphs over half-spaces are flat.
Study anisotropic flow for capillary hypersurfaces, proving new inequalities.
problem Anisotropic capillary hypersurfaces and their properties.
method Anisotropic volume-preserving mean curvature flow, new approach for strictly convex initial hypersurfaces.
result Established new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces.
In this note, we give a classification of complete anisotropic isoparametric hypersurfaces, i.e., hypersurfaces with constant anisotropic principal curvatures, in Euclidean spaces, which is in analogue with the classical case for isoparametric hypersurfaces in Euclidean spaces. On the other hand, by an example of local…
Computes Finsler geometry curvature using anisotropic connections.
problem Computing curvature tensors in Finsler geometry.
method Using affine connections to compute curvature tensors without coordinates.
result Obtained Bianchi identities and comparison of curvature tensors.
Graphs with bounded anisotropic mean curvature are regular almost everywhere.
problem Understanding the regularity of graphs with anisotropic mean curvature.
method Proving regularity for m-dimensional Lipschitz graphs with anisotropic mean curvature bounded in Lp. result Graphs with bounded anisotropic mean curvature are regular almost everywhere.
Study anisotropic capillary surfaces in a wedge using generalized Minkowski norms.
problem Understanding capillary surfaces with anisotropic forces.
method Generalized Minkowski norm on the unit sphere, new Minkowski formulae, Heintze-Karcher inequality.
result Proved an Alexandrov-type theorem in the anisotropic setting.
We study surfaces with constant anisotropic mean curvature which are invariant under a helicoidal motion. For functionals with axially symmetric Wulff shapes, we generalize the recently developed twizzler representation of Perdomo to the anisotropic case and show how all helicoidal constant anisotropic mean curvature s…
The study finds anisotropic minimal surfaces in 3-manifolds with smooth boundaries.
problem Finding smooth anisotropic minimal surfaces in closed 3-manifolds.
method Min-max construction with elliptic integrands, uniform upper bound for density ratios.
result Obtains a smooth anisotropic minimal surface in a closed 3-manifold.
Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.
problem Estimating the Morse index of anisotropic minimal surfaces.
method Local analysis of Gauss map, conformal geometric techniques applied to the Gauss map.
result Upper and lower estimates for the Morse index of anisotropic minimal surfaces.
Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.
problem Extending capillary convex body results to anisotropic setting.
method Developed theory for anisotropic capillary convex bodies in half-space and established Alexandrov-Fenchel inequality for mixed volumes.
result Established a general Alexandrov-Fenchel inequality for mixed volumes of anisotropic capillary convex bodies, weakening and extending previous results.
Higher order anisotropic superspaces are constructed as generalized vector superbundles provided with compatible nonlinear connection, distinguished connection and metric structures.
Given a positive function F on Sn which satisfies a convexity condition, for 1≤r≤n, we define for hypersurfaces in Rn+1 the r-th anisotropic mean curvature function Hr;F, a generalization of the usual r-th mean curvature function. We call a hypersurface is anisotropic mini…
The paper studies a curve flow preserving anisotropic length for convex curves, leading to a homothetic limit.
problem Anisotropic length preservation in curve deformation.
method A curve flow that maintains anisotropic length, analyzed for convex closed curves.
result Convex curves evolve to homothetic limits of Wulff shapes as time approaches infinity.
Study on special anisotropic conformal changes of conic pseudo-Finsler surfaces.
problem Exploring various anisotropic conformal transformations of conic pseudo-Finsler surfaces.
method Presented various anisotropic conformal transformations including C-anisotropic, horizontal C-anisotropic, and vertical C-anisotropic transformations. result Vertical φT-condition transformation makes every Landsberg surface Berwaldian.