Maps between non-compact surfaces can have geometric kernels under certain conditions.
problem Understanding when maps between non-compact surfaces have geometric kernels.
method Using Brown's proper fundamental group to establish sufficient conditions for geometric kernels.
result Characterization of conjugacy classes in the proper fundamental group and sufficient conditions for geometric kernels.
We present a geometric formulation of the Multiple Kernel Learning (MKL) problem. To do so, we reinterpret the problem of learning kernel weights as searching for a kernel that maximizes the minimum (kernel) distance between two convex polytopes. This interpretation combined with novel structural insights from our geom…
Survey on manifold ends with new heat kernel estimates.
problem Analyzing geometric properties on manifolds with ends.
method Constructing manifolds with ends and analyzing their heat kernel estimates.
result Found manifolds with ends that have different heat kernel estimates.
Proves upper bounds for heat kernels evolving on manifolds.
problem Bounding heat kernels on evolving manifolds.
method Logarithmic Sobolev inequalities and ultracontractivity estimates.
result Gaussian upper bounds for heat kernels are derived.
Extends heat kernel estimates for super Ricci flow.
problem Heat kernel estimates for super Ricci flow.
method Generalizes Bamler-Zhang's geometric analysis to super Ricci flow.
result Obtains Gaussian heat kernel estimates for super Ricci flow.
Local index theorem for chiral geometric operators proved using heat kernel.
problem Proving a local index theorem for geometric first-order differential operators.
method Using Gilkey's invariance theory and heat kernel techniques.
result Supertrace of heat kernel converges to Chern-Weil form.
New approximative kernels improve PDE-G-CNNs for geometric deep learning.
problem Inaccurate approximations of exact kernels in PDE-G-CNNs.
method Developed new approximative kernels that work regardless of spatial anisotropy.
result New kernels provide better error estimates and maintain reflectional symmetries.
Geometric regularisation improves statistical models by avoiding degeneracy loci.
problem Non-identifiability, singular information, and moment indeterminacy in statistical models.
method Develops the geometric regularisation of distribution-kernel pairs (T,φ) using Whitney, Thom, and Mather theorems. result Finite-dimensional weak transversality theorem for generic kernels, avoiding degeneracy strata of high codimension.
Kernel VICReg improves SSL in RKHS, capturing nonlinear structures.
problem Limited ability of existing SSL methods to handle nonlinear dependencies.
method Kernel VICReg framework in RKHS, kernelizing VICReg objectives.
result Kernel VICReg mitigates representational collapse and improves performance.
A new method for approximating softmax and Gaussian kernels with reduced error.
problem Approximating softmax and Gaussian kernels with low error.
method Simplex Random Features (SimRFs) and SimRFs+.
result SimRFs provide the smallest MSE among weight-independent geometrically-coupled PRF mechanisms.
The purpose of this paper is to extend the explicit geometric evaluation of semisimple orbital integrals for smooth kernels for the Casimir operator obtained by the first author to the case of kernels for arbitrary elements in the center of the enveloping algebra.
This work investigates a mixture of LMC and RMHMC with MMALA for geometric ergodicity.
problem Lack of geometric ergodicity study in Riemannian manifold and Lagrangian Monte Carlo methods.
method Investigates a mixture of LMC and RMHMC with MMALA to achieve geometric ergodicity.
result Demonstrates geometric ergodicity in the mixture of LMC and RMHMC with MMALA.
GeometricKernels package implements kernels for uncertain data on graphs, manifolds, and meshes.
problem Defining and computing kernels for structured data on graphs, manifolds, and meshes.
method Implementation of geometric analogs of Euclidean kernels (heat and Matérn) with automatic differentiation support.
result Ability to compute Fourier-feature-type expansions on geometric spaces.
Geometric theory connects machine learning classifiers to differential geometry.
problem Classifying data points in machine learning.
method Mapping binary classification to vector bundles and differential geometry.
result Harmonic interpolation solves RKHS interpolation problems.
A new kernel for ranked data tackles computational challenges.
problem Complex geometric structure and partial rankings make existing algorithms infeasible for real-world applications.
method Derives a graph cut kernel that combines submodular optimization and kernel-based methods.
result The graph cut kernel efficiently handles large-scale ranked data.
Study geometric bounds on generalized Ricci flow.
problem No specific problem stated; focuses on bounds.
method Analogous geometric quantities and bounds proven.
result Geometric and analytic bounds established.
Smooth bundles with rough data maintain Hodge kernel isomorphism.
problem Maintaining Hodge kernel isomorphism for smooth bundles with non-smooth geometric data.
method Analyzing nilpotent differential operators and Hodge-Dirac-type operators under perturbations of geometric data.
result Kernels of Hodge-Dirac operators remain isomorphic under uniform perturbations of geometric data.
Study shows how to reduce data needed for learning under geometric constraints.
problem Learning high-dimensional data with geometric priors.
method Spherical harmonic decompositions and kernel methods for invariance and geometric stability.
result Improvements in sample complexity by leveraging group invariance, with asymptotic behavior depending on spectral properties.
Develops kernels for matchings, overcoming computational challenges.
problem Challenges in applying kernel methods to matchings due to their discrete, non-Euclidean nature.
method Characterizes stationary kernels, introduces heat and Matérn kernel families, and develops a sub-exponential algorithm for efficient evaluation.
result Establishes novel negative results and identifies an open problem in transferring the framework to trees.
The rate of convergence of weighted kernel herding (WKH) and sequential Bayesian quadrature (SBQ), two kernel-based sampling algorithms for estimating integrals with respect to some target probability measure, is investigated. Under verifiable conditions on the chosen kernel and target measure, we establish a near-geom…
New approach interprets Nyström for kernel machines with geometric insight.
problem No comparative study over Nyström-based kernel machine approaches.
method Developed a new approach with geometric interpretation, showing equivalence to existing methods.
result Proposed approach offers insights into approximation errors and accuracy.
Proposes IIKL for preserving geometric properties of non-Euclidean data.
problem Loss of geometric information in non-Euclidean data representation.
method IIKL method builds Riemannian manifold and isometrically induces metric.
result Preserves geometric structure of original data in 3D and high-dimensional datasets.
Study Szegő kernel on non-compact CR manifolds with specific conditions.
problem Analyzing Szegő kernel on non-compact CR manifolds.
method Establish Szegő kernel asymptotic expansions on non-compact strictly pseudoconvex CR manifolds with transversal CR R-action under natural geometric conditions. result Szegő kernel asymptotic expansions established on non-compact CR manifolds.
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.
tf_geometric simplifies graph deep learning in TensorFlow.
problem Efficient graph deep learning in TensorFlow.
method Kernel libraries and infrastructures for GNNs.
result tf_geometric supports various graph tasks and provides efficient GNN models.
Paper develops methods for analyzing forms with synchronized singularities.
problem Analyzing forms with synchronized singularities.
method Exact reduction, analytic transfer, and geometric recomposition.
result Transfer of sparse domination principle to synchronized singular forms.
In this paper, we study the problem of sparse multiple kernel learning (MKL), where the goal is to efficiently learn a combination of a fixed small number of kernels from a large pool that could lead to a kernel classifier with a small prediction error. We develop an efficient algorithm based on the greedy coordinate d…
Researchers develop a generalised geometric Brownian motion for better asset pricing.
problem Irregularities in simple geometric Brownian motion for asset dynamics.
method Introduce a memory kernel to generalise GBM, derive moments and probability density functions.
result The performance of kernels in pricing options depends on option maturity and moneyness.
New summary measures reveal geometric structure in weighted measures on manifolds.
problem Lack of geometric information in standard weight-only summaries.
method Heat-kernel entropy profiles, tracking nonuniformity across scales.
result Geometric effective sample size discounts nearby or duplicate particles.
We generalize several recent results concerning the asymptotic expansions of Bergman kernels to the framework of geometric quantization and establish an asymptotic symplectic identification property. More precisely, we study the asymptotic expansion of the G-invariant Bergman kernel of the spin^c Dirac operator assoc…
We study the expressive power of kernel methods and the algorithmic feasibility of multiple kernel learning for a special rich class of kernels. Specifically, we define \emph{Euclidean kernels}, a diverse class that includes most, if not all, families of kernels studied in literature such as polynomial kernels and radi…
We introduce a new geometric approach that constructs a transition kernel of Markov chain. Our method always minimizes the average rejection rate and even reduce it to zero in many relevant cases, which cannot be achieved by conventional methods, such as the Metropolis-Hastings algorithm or the heat bath algorithm (Gib…
The paper proves various inequalities on gradient shrinking Ricci solitons.
problem Understanding geometric inequalities on gradient shrinking Ricci solitons.
method Proving multiple inequalities equivalent on complete gradient shrinking Ricci solitons.
result Various inequalities (Sobolev, logarithmic Sobolev, Schrödinger, etc.) are equivalent on gradient shrinking Ricci solitons.
Graph kernels are widely used for measuring the similarity between graphs. Many existing graph kernels, which focus on local patterns within graphs rather than their global properties, suffer from significant structure information loss when representing graphs. Some recent global graph kernels, which utilizes the align…
Study provides bounds for estimating intrinsic dimension using Gaussian kernels.
problem Estimating intrinsic dimension from data.
method Finite-sample concentration and anti-concentration bounds for Gaussian kernel sums.
result Explicit dependence on sample size, bandwidth, and geometric parameters.
We give a purely complex geometric proof of the existence of the Bergman kernel expansion. Our method provides a sharper estimate, and in the case that the metrics are real analytic, we prove that the remainder decays faster than any polynomial.
For binary classification we establish learning rates up to the order of n−1 for support vector machines (SVMs) with hinge loss and Gaussian RBF kernels. These rates are in terms of two assumptions on the considered distributions: Tsybakov's noise assumption to establish a small estimation error, and a new geometr…
New method for learning with non-Euclidean data using decomposable kernels.
problem Difficulty in using classical kernels for non-Euclidean data.
method Reproducing kernel Krein space (RKKS) methods for kernels that admit a positive decomposition.
result Invariant kernels can be used for learning in non-Euclidean spaces.
Hermann Schwarz, while studying complex analysis, introduced the geometric interpretation for the Poisson kernel in 1890. We shall see here that the geometric interpretation can be useful to develop a new approach to some old classical problems as well as to obtain several new results, mostly related to hyperbolic geom…
3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.
problem Understanding which 3-manifold groups can have convex co-compact representations.
method Analyzing representations of 3-manifold groups into projective general linear group, focusing on convex co-compactness.
result Fundamental groups of closed irreducible orientable 3-manifolds can only admit convex co-compact representations if they are geometric or hyperbolic.
Derives geometrically a description of a 3-manifold's second homotopy group.
problem Describing the second homotopy group of a 3-manifold using kernels of homomorphisms.
method Geometric derivation of Lei and Wu's result.
result Geometric derivation of the second homotopy group description.
Paper proposes a new method to learn distribution kernels via entropy maximization.
problem Challenges in applying kernel methods to distribution regression tasks.
method Proposes a novel objective for unsupervised learning of data-dependent distribution kernels based on entropy maximization.
result Demonstrates the effectiveness of the learned kernel across different modalities.
Overview of geometric analysis for manifold learning.
problem Analyzing high-dimensional data via spectral embeddings.
method Heat kernel and eigenfunctions on Riemannian manifolds.
result Uniform control of spectral embeddings on key classes of manifolds.
Support Vector Data Description (SVDD) is a machine-learning technique used for single class classification and outlier detection. SVDD formulation with kernel function provides a flexible boundary around data. The value of kernel function parameters affects the nature of the data boundary. For example, it is observed …
GDB bridges geometric states with improved accuracy and generality.
problem Challenges in predicting geometric state evolution in complex systems.
method Geometric Diffusion Bridge (GDB) framework using equivariant diffusion bridges.
result GDB surpasses existing methods in accurately bridging geometric states.
Nonnegative Matrix Factorization (NMF) has been continuously evolving in several areas like pattern recognition and information retrieval methods. It factorizes a matrix into a product of 2 low-rank non-negative matrices that will define parts-based, and linear representation of nonnegative data. Recently, Graph regula…
Study heat kernel on quaternionic contact manifolds, finding linear dependence of coefficients on curvature.
problem Analyzing heat kernel on quaternionic contact manifolds.
method Explicit computation of heat kernel coefficients and dependence on curvature.
result Second coefficient of heat kernel's small time asymptotics depends linearly on the qc scalar curvature.
Novel heat flow estimates on ALE manifolds for Schrödinger operators.
problem Estimating heat flows on ALE manifolds with non-trivial L2-kernel. method Combining Fredholm theory for Dirac type operators and heat kernel advances.
result Established Lp−Lq decay estimates for heat flows.