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We introduce a data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves…
Traditionally, kernel methods rely on the representer theorem which states that the solution to a learning problem is obtained as a linear combination of the data mapped into the reproducing kernel Hilbert space (RKHS). While elegant from…
Kernel-based clustering algorithms have the ability to capture the non-linear structure in real world data. Among various kernel-based clustering algorithms, kernel k-means has gained popularity due to its simple iterative nature and ease…
We provide the first mathematically complete derivation of the Nystr\"om method for low-rank approximation of indefinite kernels and propose an efficient method for finding an approximate eigendecomposition of such kernel matrices. Building…
Data-driven control algorithms use observations of system dynamics to construct an implicit model for the purpose of control. However, in practice, data-driven techniques often require excessive sample sizes, which may be infeasible in…
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…
We prove rates of convergence in the statistical sense for kernel-based least squares regression using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is directly related…
In this paper we introduce a reproducing kernel Hilbert space defined on $\mathbb{R}^{d+1}$ as the tensor product of a reproducing kernel defined on the unit sphere $\mathbb{S}^{d}$ in $\mathbb{R}^{d+1}$ and a reproducing kernel defined on…
The aim of this study is to present a good modernistic strategy for solving some well-known classes of Lane-Emden type singular differential equations. The proposed approach is based on the reproducing kernel Hilbert space (RKHS) and…
We present several generative and predictive algorithms based on the RKHS (reproducing kernel Hilbert spaces) methodology, which, most importantly, are scale up efficiently with large datasets or high-dimensional data. It is well recognized…
For the past 30 years or so, machine learning has stimulated a great deal of research in the study of approximation capabilities (expressive power) of a multitude of processes, such as approximation by shallow or deep neural networks,…
This paper studies the probabilistic function approximation problem over reproducing kernel Hilbert spaces. We show the existence and uniqueness of the optimizer under mild assumptions. Furthermore, we generalize the celebrated representer…
Face Recognition (FR) has been the interest to several researchers over the past few decades due to its passive nature of biometric authentication. Despite high accuracy achieved by face recognition algorithms under controlled conditions,…
Motivated by the growing interest in representation learning approaches that uncover the latent structure of high-dimensional data, this work proposes new algorithms for reconstruction-based manifold learning within Reproducing-Kernel…
Reachability computations that rely on learned or estimated models require calibration in order to uphold confidence about their guarantees. Calibration generally involves sampling scenarios inside the reachable set. However, producing…
Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the "native" Hilbert space $\calh$ in which they are reproducing. Continuous kernels on compact domains have an expansion into…
The distribution regression problem encompasses many important statistics and machine learning tasks, and arises in a large range of applications. Among various existing approaches to tackle this problem, kernel methods have become a method…
The performance of adaptive estimators that employ embedding in reproducing kernel Hilbert spaces (RKHS) depends on the choice of the location of basis kernel centers. Parameter convergence and error approximation rates depend on where and…
Random feature approximation is arguably one of the most popular techniques to speed up kernel methods in large scale algorithms and provides a theoretical approach to the analysis of deep neural networks. We analyze generalization…
Kernel-based methods enjoy powerful generalization capabilities in handling a variety of learning tasks. When such methods are provided with sufficient training data, broadly-applicable classes of nonlinear functions can be approximated…