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Over the last decade, kernel methods for nonlinear processing have successfully been used in the machine learning community. The primary mathematical tool employed in these methods is the notion of the Reproducing Kernel Hilbert Space.…
Spectral kernel methods are techniques for transforming data into a coordinate system that efficiently reveals the geometric structure - in particular, the "connectivity" - of the data. These methods depend on certain tuning parameters. We…
Quadrature by Expansion (QBX) is a quadrature method for approximating the value of the singular integrals encountered in the evaluation of layer potentials. It exploits the smoothness of the layer potential by forming locally-valid…
This paper describes an adaptive method in continuous time for the estimation of external fields by a team of $N$ agents. The agents $i$ each explore subdomains $\Omega^i$ of a bounded subset of interest $\Omega\subset X := \mathbb{R}^d$.…
Kernel approximation methods create explicit, low-dimensional kernel feature maps to deal with the high computational and memory complexity of standard techniques. This work studies a supervised kernel learning methodology to optimize such…
Quantum kernel methods are a promising branch of quantum machine learning, yet their effectiveness on diverse, high-dimensional, real-world data remains unverified. Current research has largely been limited to low-dimensional or synthetic…
Kernelization is an important tool in parameterized algorithmics. Given an input instance accompanied by a parameter, the goal is to compute in polynomial time an equivalent instance of the same problem such that the size of the reduced…
Kernel methods are versatile tools for function approximation and surrogate modeling. In particular, greedy techniques offer computational efficiency and reliability through inherent sparsity and provable convergence. Inspired by the…
The field of quantum machine learning is a promising way to lead to a revolution in intelligent data processing methods. In this way, a hybrid learning method based on classic kernel methods is proposed. This proposal also requires the…
By removing irrelevant and redundant features, feature selection aims to find a good representation of the original features. With the prevalence of unlabeled data, unsupervised feature selection has been proven effective in alleviating the…
This paper presents a method for the approximation of harmonic potentials that combines downward continuation of globally available data on a sphere $\Omega_R$ of radius $R$ (e.g., a satellite's orbit) with locally available data on a…
An efficient computational approach for optimal reconstructing parameters of binary-type physical properties for models in biomedical applications is developed and validated. The methodology includes gradient-based multiscale optimization…
This paper presents a two-phase method for learning interaction kernels of stochastic many-particle systems. After transforming stochastic trajectories of every particle into the particle density function by the kernel density estimation…
In this paper, we develop a quadrature framework for large-scale kernel machines via a numerical integration representation. Considering that the integration domain and measure of typical kernels, e.g., Gaussian kernels, arc-cosine kernels,…
The random feature (RF) approach is a well-established and efficient tool for scalable kernel methods, but existing literature has primarily focused on kernel ridge regression with random features (KRR-RF), which has limitations in handling…
Kernelization algorithms, usually a preprocessing step before other more traditional algorithms, are very special in the sense that they return (reduced) instances, instead of final results. This characteristic excludes the freedom of…
It is well known that the quasi-optimality of the Galerkin finite element method for the Helmholtz equation is dependent on the mesh size and the wave-number. In the literature, different criteria have been proposed to ensure uniform…
Kernel methods provide a principled approach to nonparametric learning. While their basic implementations scale poorly to large problems, recent advances showed that approximate solvers can efficiently handle massive datasets. A shortcoming…
This paper presents an iterative method suitable for inverting semilinear problems which are important kernels in many numerical applications. The primary idea is to employ a parametrization that is able to reduce semilinear problems into…
A novel algorithm to solve the quadratic programming problem over ellipsoids is proposed. This is achieved by splitting the problem into two optimisation sub-problems, quadratic programming over a sphere and orthogonal projection. Next, an…