arXiv research

Unified framework for constructing RKBSs with various norms and kernels.

Proposes a method to learn a low-rank kernel matrix for graph-based clustering.

Kernel for multi-parameter persistent homology connects TDA with ML.

Improved kernel quadrature with convex weights using subsampling.

Develops method to construct Lie algebra weight system kernel using Vogel algebra.

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…

Unified framework for constructing kernels for transport equations and Koopman eigenfunctions.

New kernels from neural networks show better performance than traditional methods.

Kernel means are frequently used to represent probability distributions in machine learning problems. In particular, the well known kernel density estimator and the kernel mean embedding both have the form of a kernel mean. Unfortunately, kernel means are faced with scalability issues. A single point evaluation of the …

This work explores variably scaled kernels to improve non-stationary Gaussian processes.

The problem of estimating the kernel mean in a reproducing kernel Hilbert space (RKHS) is central to kernel methods in that it is used by classical approaches (e.g., when centering a kernel PCA matrix), and it also forms the core inference step of modern kernel methods (e.g., kernel-based non-parametric tests) that rel…

Maps embed manifolds using heat kernels of connection Laplacian.

The heat kernel for the Cauchy-Riemann subLaplacian on S(2n+1) is derived in a manner which is completely analogous to the classical derivation of elliptic heat kernels. This suggests that the classical hamiltonian construction of elliptic heat kernels, with appropriate modifications, does yield heat kernels for subell…

The paper constructs optimal confidence bands for kernel gradient flow estimators.

Kernel methods are widespread in machine learning; however, they are limited by the quadratic complexity of the construction, application, and storage of kernel matrices. Low-rank matrix approximation algorithms are widely used to address this problem and reduce the arithmetic and storage cost. However, we observed tha…

We consider second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold without boundary, working without the assumption of Laplace-like principal part $-\N^μ\N_μ$. Our objective is to obtain information on the asymptotic expansions of the corresponding r…

No-trick kernel adaptive filtering uses deterministic features for scalability and robustness.

Quantum kernels can be efficiently embedded into classical feature spaces.

We analyze the size of the dictionary constructed from online kernel sparsification, using a novel formula that expresses the expected determinant of the kernel Gram matrix in terms of the eigenvalues of the covariance operator. Using this formula, we are able to connect the cardinality of the dictionary with the eigen…

Forest-based kernels improve on existing methods for high-dimensional testing.

Develops robust persistence diagrams using kernel methods.

KernelNet learns data-dependent kernels for deep generative models.

New wavelet frames constructed from reproducing kernels for continuous and discrete domains.

Kernel-based SSL creates useful representations without labels.

We investigate iterated compositions of weighted sums of Gaussian kernels and provide an interpretation of the construction that shows some similarities with the architectures of deep neural networks. On the theoretical side, we show that these kernels are universal and that SVMs using these kernels are universally con…

We construct near-optimal coresets for kernel density estimates for points in $\mathbb{R}^d$ when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size $O(\sqrt{d}/\varepsilon\cdot \sqrt{\log 1/\varepsilon} )$, and we show a near-matching lower bound of size $Ω(\min\…

Kernel methods linked to feature subspaces and maximal correlation kernels.

Most kernel-based methods, such as kernel or Gaussian process regression, kernel PCA, ICA, or $k$-means clustering, do not scale to large datasets, because constructing and storing the kernel matrix $\mathbf{K}_n$ requires at least $\mathcal{O}(n^2)$ time and space for $n$ samples. Recent works show that sampling point…

Kernel Flows approximates unknown functions using numerical kernel selection.

Constructs index for elliptic operators using rapidly decaying kernels.

We develop Fourier methods to expand translation-invariant kernels.

A new method warps inputs to learn nonstationary kernels efficiently.

New approach to supervised learning in RKHS and vvRKHS using $C^*$-algebras.

We consider a heat kernel approach for the development of stochastic pricing kernels. The kernels are constructed by positive propagators, which are driven by time-inhomogeneous Markov processes. We multiply such a propagator with a positive, time-dependent and decreasing weight function, and integrate the product over…

We present Random Partition Kernels, a new class of kernels derived by demonstrating a natural connection between random partitions of objects and kernels between those objects. We show how the construction can be used to create kernels from methods that would not normally be viewed as random partitions, such as Random…

We lay theoretical foundations for new database release mechanisms that allow third-parties to construct consistent estimators of population statistics, while ensuring that the privacy of each individual contributing to the database is protected. The proposed framework rests on two main ideas. First, releasing (an esti…

Positive-definite kernel functions are fundamental elements of kernel methods and Gaussian processes. A well-known construction of such functions comes from Bochner's characterization, which connects a positive-definite function with a probability distribution. Another construction, which appears to have attracted less…

New estimator reduces kernel mean estimation error.

In the framework of path integral the evolution operator kernel for the Merton-Garman Hamiltonian is constructed. Based on this kernel option formula is obtained, which generalizes the well-known Black-Scholes result. Possible approximation numerical schemes for path integral calculations are proposed.

We construct a canonical correspondence from a wide class of reproducing kernels on infinite-dimensional Hermitian vector bundles to linear connections on these bundles. The linear connection in question is obtained through a pull-back operation involving the tautological universal bundle and the classifying morphism o…

Many real world graphs, such as the graphs of molecules, exhibit structure at multiple different scales, but most existing kernels between graphs are either purely local or purely global in character. In contrast, by building a hierarchy of nested subgraphs, the Multiscale Laplacian Graph kernels (MLG kernels) that we …

We introduce the Mondrian kernel, a fast random feature approximation to the Laplace kernel. It is suitable for both batch and online learning, and admits a fast kernel-width-selection procedure as the random features can be re-used efficiently for all kernel widths. The features are constructed by sampling trees via a…

Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the properties that (1) B possesses an l1 norm in the sense that it is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous lin…

Interest in multioutput kernel methods is increasing, whether under the guise of multitask learning, multisensor networks or structured output data. From the Gaussian process perspective a multioutput Mercer kernel is a covariance function over correlated output functions. One way of constructing such kernels is based …

Gaussian processes adapted for Riemannian manifolds using gauge-independent kernels.

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…

New framework estimates eigenvalues of kernel matrices without full matrix construction.

In this work, we propose the kernel Pitman-Yor process (KPYP) for nonparametric clustering of data with general spatial or temporal interdependencies. The KPYP is constructed by first introducing an infinite sequence of random locations. Then, based on the stick-breaking construction of the Pitman-Yor process, we defin…