arXiv research

Optimal Poincaré constant estimates on manifolds with ends.

The paper studies Lipschitz bounds for integral kernels under differentiability assumptions.

Kernel interpolation is inconsistent for norms with smoothness above a constant.

Study on kernel regression risk in high dimensions using Pinsker bound.

Study classifies solutions to specific equations on half-space and ball.

Wide neural networks become linear, with constant tangent kernel, due to Hessian scaling.

Improved guarantees for misspecified kernelized bandit optimization.

The paper studies estimating the normalizing constant using queries to a black-box function in RKHS.

We study the construction of coresets for kernel density estimates. That is we show how to approximate the kernel density estimate described by a large point set with another kernel density estimate with a much smaller point set. For characteristic kernels (including Gaussian and Laplace kernels), our approximation pre…

Researchers found a Weyl law for Liouville quantum gravity eigenvalues.

We show that minimum-norm interpolation in the Reproducing Kernel Hilbert Space corresponding to the Laplace kernel is not consistent if input dimension is constant. The lower bound holds for any choice of kernel bandwidth, even if selected based on data. The result supports the empirical observation that minimum-norm …

The paper examines how adversarial training and noise affect neural network performance.

This article shows that if the negative part of Ricci curvature lies in the Kato class, the heat kernel satisfies a Li-Yau type estimate. Additionally, using the resulting heat kernel bound, we show that the obtained heat kernel estimate leads to bounds on the first Betti number only depending on the Kato constant.

In this paper, we systematically study the heat kernel of the Ricci flows induced by Ricci shrinkers. We develop several estimates which are much sharper than their counterparts in general closed Ricci flows. Many classical results, including the optimal Logarithmic Sobolev constant estimate, the Sobolev constant estim…

Estimates Bergman kernels on Kähler manifolds with Ricci bounds.

We prove Gray--Moser stability theorems for complementary pairs of forms of constant class defining symplectic pairs, contact-symplectic pairs and contact pairs. We also consider the case of contact-symplectic and contact-contact structures, in which the constant class condition on a one-form is replaced by the conditi…

Without using the $L^2$ extension theorem, we provide a new proof of the equality part in Suita's conjecture, which states that for any open Riemann surface admitting a Green's function, the Bergman kernel and the logarithmic capacity coincide at one point if and only if the surface is biholomorphic to a disc possibly …

A faster graph kernel using optical random features.

The paper proves the existence of a special type of metric on complex manifolds.

We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the $L^2$ Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions.

Sharp fractional Sobolev inequalities on closed manifolds identified.

This work analyzes when contrastive models are close to PCA or kernel methods.

Bayesian Gaussian Processes improve exoplanet transit and Hubble constant inference.

Most state-of-the-art graph kernels only take local graph properties into account, i.e., the kernel is computed with regard to properties of the neighborhood of vertices or other small substructures. On the other hand, kernels that do take global graph propertiesinto account may not scale well to large graph databases.…

We study the subelliptic heat kernel of the sub-Laplacian on a 2n+1-dimensional anti-de Sitter space H2n+1 which also appears as a model space of a CR Sasakian manifold with constant negative sectional curvature. In particular we obtain an explicit and geometrically meaningful formula for the subelliptic heat kernel. T…

Paper analyzes SGD in kernel regression, showing it outperforms offline methods.

We prove the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type: $|\nabla P_t f| \le K P_t(|\nabla f|)$ where $P_t$ is the heat semigroup corresponding to the sublaplacian on $G$, $\nabla$ is the subelliptic gradient, and $K$ is a constant. This extends a result of H.-…

Paper studies kernel hyperparameters for clustering, proposing an efficient search method.

In this paper, we solve the optimal constant problem in the setting of Ohsawa's generalized $L^{2}$ extension theorem. As applications, we prove a conjecture of Ohsawa and the extended Suita conjecture, we also establish some relations between Bergman kernel and logarithmic capacity on compact and open Riemann surfaces…

This paper uses machine learning to select kernels for machine learning models on various devices.

Analog arrays are a promising upcoming hardware technology with the potential to drastically speed up deep learning. Their main advantage is that they compute matrix-vector products in constant time, irrespective of the size of the matrix. However, early convolution layers in ConvNets map very unfavorably onto analog a…

This work challenges the Neural Tangent Kernel's role in overparameterized neural networks, especially with large width and depth.

Let $(M,\,g)$ be a Poincar$\acute{\text{e}}$-Einstein manifold with a smooth defining function. In this note, we prove that there are infinitely many asymptotically hyperbolic metrics with constant $Q$-curvature in the conformal class of an asymptotically hyperbolic metric close enough to $g$. These metrics are paramet…

Study finds the number of modes in Gaussian kernel density estimators scales with sqrt(β log β).

We establish refinements of the classical Kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first-order linear differential operator. Our main result is a general expression which gives the value of the constants appearing in the refined inequalities. These consta…

Changing kernel bandwidth during training improves kernel regression performance.

Enhances random forests by smoothing predictions for better performance.

We estimate the heat kernel on a closed Riemannian manifold $M$, with $dim(M)\geq 3$, evolving under the Ricci-harmonic map flow and the result depends on some constants arising from a Sobolev imbedding theorem. In a special case, when the scalar curvature satisfies a certain natural inequality, we obtain, as a corolla…

The study proves that sets with constant nonlocal curvature are composed of equal balls under certain conditions.

The paper examines the optimality of kernel methods in high-dimensional clustering.

We consider the heat equation associated with a class of second order hypoelliptic Hörmander operators with constant second order term and linear drift. We describe the possible small time heat kernel expansion on the diagonal giving a geometric characterization of the coefficients in terms of the divergence of the dri…

The paper explores the identifiability and interpretability of Gaussian process models using different kernel structures.

We derive a local Gaussian upper bound for the $f$-heat kernel on complete smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-Émery Ricci curvature, which generalizes the classic Li-Yau estimate. As applications, we obtain a sharp $L_f^1$-Liouville theorem for $f$-subharmonic functions and an $L_f^1$-u…

We study the concentration of random kernel matrices around their mean. We derive nonasymptotic exponential concentration inequalities for Lipschitz kernels assuming that the data points are independent draws from a class of multivariate distributions on $\mathbb R^d$, including the strongly log-concave distributions u…

Positive definite kernels and their associated Reproducing Kernel Hilbert Spaces provide a mathematically compelling and practically competitive framework for learning from data. In this paper we take the approximation theory point of view to explore various aspects of smooth kernels related to their inferential proper…

Improved kernel Stein discrepancy for large-scale data.

Study shows convergence of cscK surfaces in Hilbert scheme.

New research sets the minimax lower bound for KSD estimation at sqrt(n).