Related papers: Operator-Valued Bochner Theorem, Fourier Feature M…
Random features have been introduced to scale up kernel methods via randomization techniques. In particular, random Fourier features and orthogonal random features were used to approximate the popular Gaussian kernel. Random Fourier…
This paper presents novel generalization bounds for vector-valued neural networks and deep kernel methods, focusing on multi-task learning through an operator-theoretic framework. Our key development lies in strategically combining a…
Kernel methods approximate nonlinear maps in a data-driven manner by projecting the target map onto a finite-dimensional Hilbert space called the solution space. Traditionally, this space is a subspace of a fixed ambient reproducing kernel…
Quantum machine learning (QML) is the spearhead of quantum computer applications. In particular, quantum neural networks (QNN) are actively studied as the method that works both in near-term quantum computers and fault-tolerant quantum…
We analyze in this paper a random feature map based on a theory of invariance I-theory introduced recently. More specifically, a group invariant signal signature is obtained through cumulative distributions of group transformed random…
Approximating non-linear kernels using feature maps has gained a lot of interest in recent years due to applications in reducing training and testing times of SVM classifiers and other kernel based learning algorithms. We extend this line…
We propose a principled method for kernel learning, which relies on a Fourier-analytic characterization of translation-invariant or rotation-invariant kernels. Our method produces a sequence of feature maps, iteratively refining the SVM…
It is shown that a positive (bounded linear) operator on a Hilbert space with trivial kernel is unitarily equivalent to a Hankel operator that satisfies double positivity condition if and only if it is non-invertible and has simple spectrum…
The kernel trick in supervised learning signifies transformations of an inner product by a feature map, which then restructures training data in a larger Hilbert space according to an endowed inner product. A quantum feature map corresponds…
This paper introduces the Kernel Neural Operator (KNO), a provably convergent operator-learning architecture that utilizes compositions of deep kernel-based integral operators for function-space approximation of operators (maps from…
The use of covariance kernels is ubiquitous in the field of spatial statistics. Kernels allow data to be mapped into high-dimensional feature spaces and can thus extend simple linear additive methods to nonlinear methods with higher order…
Recent advances in operator learning theory have improved our knowledge about learning maps between infinite dimensional spaces. However, for large-scale engineering problems such as concurrent multiscale simulation for mechanical…
A method for analyzing the feature map for the kernel-based quantum classifier is developed; that is, we give a general formula for computing a lower bound of the exact training accuracy, which helps us to see whether the selected feature…
Nonlinear kernels can be approximated using finite-dimensional feature maps for efficient risk minimization. Due to the inherent trade-off between the dimension of the (mapped) feature space and the approximation accuracy, the key problem…
Kernel methods represent one of the most powerful tools in machine learning to tackle problems expressed in terms of function values and derivatives due to their capability to represent and model complex relations. While these methods show…
In a general context of positive definite kernels $k$, we develop tools and algorithms for sampling in reproducing kernel Hilbert space $\mathscr{H}$ (RKHS). With reference to these RKHSs, our results allow inference from samples; more…
We consider the problem of improving kernel approximation via randomized feature maps. These maps arise as Monte Carlo approximation to integral representations of kernel functions and scale up kernel methods for larger datasets. Based on…
A Positive Operator Valued Measure (POVM) is a map $F:\mathcal{B}(X)\to\mathcal{L}_s^+(\mathcal{H})$ from the Borel $\sigma$-algebra of a topological space $X$ to the space of positive self-adjoint operators on a Hilbert space…
Analyzing the structure of sampled features from an input data distribution is challenging when constrained by limited measurements in both the number of inputs and features. Traditional approaches often rely on the eigenvalue spectrum of…
Kernel methods are among the most popular techniques in machine learning. From a frequentist/discriminative perspective they play a central role in regularization theory as they provide a natural choice for the hypotheses space and the…