Related papers: Borel Kernels and their Approximation, Categorical…

We present a selection theorem for domains in $\mathbb{C}^n$, $n\ge 1$, which states that any tamed sequence of pointed connected open subsets admits a subsequence convergent to its own kernel in the sense of Carath\'eodory. Not only is…

Recent advances in machine learning have led to increased interest in reproducing kernel Banach spaces (RKBS) as a more general framework that extends beyond reproducing kernel Hilbert spaces (RKHS). These works have resulted in the…

We present simple, user-friendly bounds for the expected operator norm of a random kernel matrix under general conditions on the kernel function $k(\cdot,\cdot)$. Our approach uses decoupling results for U-statistics and the non-commutative…

Approximate Bayesian computation has emerged as a standard computational tool when dealing with the increasingly common scenario of completely intractable likelihood functions in Bayesian inference. We show that many common Markov chain…

This paper, broadly speaking, covers the use of randomness in two main areas: low-rank approximation and kernel methods. Low-rank approximation is very important in numerical linear algebra. Many applications depend on matrix decomposition…

This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problem instances. The paper…

Random feature approximation is arguably one of the most widely used techniques for kernel methods in large-scale learning algorithms. In this work, we analyze the generalization properties of random feature methods, extending previous…

Kernel ridge regression is used to approximate the kinetic energy of non-interacting fermions in a one-dimensional box as a functional of their density. The properties of different kernels and methods of cross-validation are explored, and…

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…

We construct a family of right coideal subalgebras of quantum groups, which have the property that all irreducible representations are one-dimensional, and which are maximal with this property. The obvious examples for this are the standard…

This paper introduces a general method to approximate the convolution of an arbitrary program with a Gaussian kernel. This process has the effect of smoothing out a program. Our compiler framework models intermediate values in the program…

It was demonstrated in earlier work that, by approximating its range kernel using shiftable functions, the non-linear bilateral filter can be computed using a series of fast convolutions. Previous approaches based on shiftable approximation…

Natural language processing often involves computations with semantic or syntactic graphs to facilitate sophisticated reasoning based on structural relationships. While convolution kernels provide a powerful tool for comparing graph…

We introduce a distance between kernels based on the Wasserstein distances between their values, study its properties, and prove that it is a metric on an appropriately defined space of kernels. We also relate it to various modes of…

These notes provide a self-contained introduction to kernel methods and their geometric foundations in machine learning. Starting from the construction of Hilbert spaces, we develop the theory of positive definite kernels, reproducing…

Kernel mean embeddings -- integrals of a kernel with respect to a probability distribution -- are essential in Bayesian quadrature, but also widely used in other computational tools for numerical integration or for statistical inference…

In this paper, we develop category theory of Markov kernels to study categorical aspects of Bayesian inversions. As a result, we present a unified model for Bayesian supervised learning, encompassing Bayesian density estimation. We…

We provide a categorical proof of convergence for martingales and backward martingales in mean, using enriched category theory. The enrichment we use is in topological spaces, with their canonical closed monoidal structure, which encodes a…

We revisit Rahimi and Recht (2007)'s kernel random Fourier features (RFF) method through the lens of the PAC-Bayesian theory. While the primary goal of RFF is to approximate a kernel, we look at the Fourier transform as a prior distribution…

It has been known in potential theory that, for some kernels matrices corresponding to well-separated point sets, fast analytical low-rank approximation can be achieved via the use of proxy points. This proxy point method gives a…