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A Hilbert space embedding of a distribution---in short, a kernel mean embedding---has recently emerged as a powerful tool for machine learning and inference. The basic idea behind this framework is to map distributions into a reproducing…
Learning the kernel functions used in kernel methods has been a vastly explored area in machine learning. It is now widely accepted that to obtain 'good' performance, learning a kernel function is the key challenge. In this work we focus on…
In this work, we consider the problem of learning nonlinear operators that correspond to discrete-time nonlinear dynamical systems with inputs. Given an initial state and a finite input trajectory, such operators yield a finite output…
We consider the regret minimization problem in reinforcement learning (RL) in the episodic setting. In many real-world RL environments, the state and action spaces are continuous or very large. Existing approaches establish regret…
Data similarity is a key concept in many data-driven applications. Many algorithms are sensitive to similarity measures. To tackle this fundamental problem, automatically learning of similarity information from data via self-expression has…
We investigate a recently proposed family of positive-definite kernels that mimic the computation in large neural networks. We examine the properties of these kernels using tools from differential geometry; specifically, we analyze the…
We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to…
Learning kernels in operators from data lies at the intersection of inverse problems and statistical learning, providing a powerful framework for capturing non-local dependencies in function spaces and high-dimensional settings. In contrast…
Depth measures have gained popularity in the statistical literature for defining level sets in complex data structures like multivariate data, functional data, and graphs. Despite their versatility, integrating depth measures into…
Generalization beyond a training dataset is a main goal of machine learning, but theoretical understanding of generalization remains an open problem for many models. The need for a new theory is exacerbated by recent observations in deep…
We propose a new technique for constructing low-rank approximations of matrices that arise in kernel methods for machine learning. Our approach pairs a novel automatically constructed analytic expansion of the underlying kernel function…
Approximate Bayesian Computation (ABC) enables statistical inference in simulator-based models whose likelihoods are difficult to calculate but easy to simulate from. ABC constructs a kernel-type approximation to the posterior distribution…
The theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and…
Performing exact posterior inference in complex generative models is often difficult or impossible due to an expensive to evaluate or intractable likelihood function. Approximate Bayesian computation (ABC) is an inference framework that…
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…
In spatial statistics and machine learning, the kernel matrix plays a pivotal role in prediction, classification, and maximum likelihood estimation. A thorough examination reveals that for large sample sizes, the kernel matrix becomes…
An important feature of kernel mean embeddings (KME) is that the rate of convergence of the empirical KME to the true distribution KME can be bounded independently of the dimension of the space, properties of the distribution and smoothness…
This article introduces and studies the tight approximation property, a property of algebraic varieties defined over the function field of a complex or real curve that refines the weak approximation property (and the known cohomological…
The search for the optimal shape parameter for Radial Basis Function (RBF) kernel approximation has been an outstanding research problem for decades. In this work, we establish a theoretical framework for this problem by leveraging a…
In this paper, we show that the approximation of high-dimensional functions, which are effectively low-dimensional, does not suffer from the curse of dimensionality. This is shown first in a general reproducing kernel Hilbert space set-up…