arXiv research

The paper develops heat kernel comparison theorems and applies them to spectral geometry.

The paper compares heat kernels on manifolds with Robin boundary conditions.

We give a version of the comparison principle from pluripotential theory where the Monge-Ampère measure is replaced by the Bergman kernel and use it to derive a maximum principle

Study Brownian motions and heat kernel bounds on Kähler and quaternion Kähler manifolds.

Unified view on random walk and Weisfeiler-Leman kernels, improving accuracy.

We propose new positive definite kernels for permutations. First we introduce a weighted version of the Kendall kernel, which allows to weight unequally the contributions of different item pairs in the permutations depending on their ranks. Like the Kendall kernel, we show that the weighted version is invariant to rela…

MKA incorporates manifold geometry into kernel alignment for more robust representation comparison.

Kernel testing compares cell states in single-cell data.

Paper compares Bergman kernel and Masur-Veech measure on Teichmüller space.

We consider the problem of metric learning subject to a set of constraints on relative-distance comparisons between the data items. Such constraints are meant to reflect side-information that is not expressed directly in the feature vectors of the data items. The relative-distance constraints used in this work are part…

Deep neural kernels and Laplace kernel have equivalent RKHS on spheres.

Bayesian Gaussian Processes improve exoplanet transit and Hubble constant inference.

Given only information in the form of similarity triplets "Object A is more similar to object B than to object C" about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a low-dimensional Euclidean embedding of the data set that reflects the given similar…

Graph kernels assess graph similarity for various applications.

A new graph kernel uses LCS and Wasserstein distance for better graph comparisons.

DEOT method compares distributions across agents with privacy and efficiency.

The Nyström methods have been popular techniques for scalable kernel based learning. They approximate explicit, low-dimensional feature mappings for kernel functions from the pairwise comparisons with the training data. However, Nyström methods are generally applied without the supervision provided by the training labe…

Gaussian kernels on complex manifolds are never positive definite.

We propose a new graph kernel for graph classification and comparison using Ollivier Ricci curvature. The Ricci curvature of an edge in a graph describes the connectivity in the local neighborhood. An edge in a densely connected neighborhood has positive curvature and an edge serving as a local bridge has negative curv…

We demonstrate that distributed block coordinate descent can quickly solve kernel regression and classification problems with millions of data points. Armed with this capability, we conduct a thorough comparison between the full kernel, the Nyström method, and random features on three large classification tasks from va…

Data analysis require a pairwise proximity measure over objects. Recent work has extended this to situations where the distance information between objects is given as comparison results of distances between three objects (triplets). Humans find the comparison tasks much easier than the exact distance computation and s…

This paper describes a new method for low rank kernel approximation called IKA. The main advantage of IKA is that it produces a function $ψ(x)$ defined as a linear combination of arbitrarily chosen functions. In contrast the approximation produced by Nyström method is a linear combination of kernel evaluations. The pro…

Standard kernels such as Matérn or RBF kernels only encode simple monotonic dependencies within the input space. Spectral mixture kernels have been proposed as general-purpose, flexible kernels for learning and discovering more complicated patterns in the data. Spectral mixture kernels have recently been generalized in…

New random feature maps for Laplacian and related kernels.

Proposes a TS approach for Bayesian optimization with preferential feedback.

Support Vector Data Description (SVDD) provides a useful approach to construct a description of multivariate data for single-class classification and outlier detection with various practical applications. Gaussian kernel used in SVDD formulation allows flexible data description defined by observations designated as sup…

Sharp heat kernel estimates on manifolds lead to solutions of the Parabolic Anderson model.

Identifies a gradient flow to solve kernel learning problems with noise reduction.

DHGAK aligns substructures for better graph kernel performance.

The paper analyzes the statistical cost of tuning kernel hyperparameters in robust regression.

New inequalities for spectral zeta kernels on spheres and manifolds.

We introduce a novel boosting algorithm called `KTBoost' which combines kernel boosting and tree boosting. In each boosting iteration, the algorithm adds either a regression tree or reproducing kernel Hilbert space (RKHS) regression function to the ensemble of base learners. Intuitively, the idea is that discontinuous …

A new metric CKCE improves model calibration comparison.

This paper establishes a kernel-based framework for reconstructing data on manifolds, tailored to fit the dynamic-(d)MRI-data recovery problem. The proposed methodology exploits simple tangent-space geometries of manifolds in reproducing kernel Hilbert spaces and follows classical kernel-approximation arguments to form…

Develops interpretable low-dimensional kernels with conic discriminant functions.

The NNGP kernel's predictions closely match those of the Matern kernel under certain conditions.

In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirichlet $p$-Laplacian ($1<p<\infty$) obtained by Matei [A.-M. Matei, First eigenvalue for the $p$-Laplace operator, Nonlinear Anal. TMA 39 (8) (2000) 1051--1068] and Takeuchi [H. Takeuchi, On the first eigenvalue of the $p$-Laplacian …

The study analyzes prediction errors in systems with memory kernels, providing bounds and stability results.

We present novel graph kernels for graphs with node and edge labels that have ordered neighborhoods, i.e. when neighbor nodes follow an order. Graphs with ordered neighborhoods are a natural data representation for evolving graphs where edges are created over time, which induces an order. Combining convolutional subgra…

Study compares exponential and power-law kernels in modeling high-frequency trading data.

Let $(X,d,μ)$ be a $RCD^\ast(K, N)$ space with $K\in \mathbb{R}$ and $N\in [1,\infty]$. For $N\in [1,\infty)$, we derive the upper and lower bounds of the heat kernel on $(X,d,μ)$ by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in t…

This paper proposes a novel kernel approach to linear dimension reduction for supervised learning. The purpose of the dimension reduction is to find directions in the input space to explain the output as effectively as possible. The proposed method uses an estimator for the gradient of regression function, based on the…

Improved sample efficiency with normalized RBF kernels in neural networks.

This work studies nonnegativity-preserving kernels for stochastic equations and their applications.

This paper introduces an approach for detecting differences in the first-order structures of spatial point patterns. The proposed approach leverages the kernel mean embedding in a novel way by introducing its approximate version tailored to spatial point processes. While the original embedding is infinite-dimensional a…

Multi-output Gaussian processes (MOGPs) are an extension of Gaussian Processes (GPs) for predicting multiple output variables (also called channels, tasks) simultaneously. In this paper we use the convolution theorem to design a new kernel for MOGPs, by modeling cross channel dependencies through cross convolution of t…

A new noise model for preferential Bayesian optimization using user anchors.

We illustrate relationships between classical kernel-based dimensionality reduction techniques and eigendecompositions of empirical estimates of reproducing kernel Hilbert space (RKHS) operators associated with dynamical systems. In particular, we show that kernel canonical correlation analysis (CCA) can be interpreted…