Related papers: An RKHS-Based Semiparametric Approach to Nonlinear…
In many scientific disciplines structures in high-dimensional data have to be found, e.g., in stellar spectra, in genome data, or in face recognition tasks. In this work we present a novel approach to non-linear dimensionality reduction. It…
Current deep image super-resolution (SR) approaches aim to restore high-resolution images from down-sampled images or by assuming degradation from simple Gaussian kernels and additive noises. However, these techniques only assume crude…
We characterize the solution of a broad class of convex optimization problems that address the reconstruction of a function from a finite number of linear measurements. The underlying hypothesis is that the solution is decomposable as a…
Latent variable models represent a useful tool for the analysis of complex data when the constructs of interest are not observable. A problem related to these models is that the integrals involved in the likelihood function cannot be solved…
This study debuts a new spline dimensional decomposition (SDD) for uncertainty quantification analysis of high-dimensional functions, including those endowed with high nonlinearity and nonsmoothness, if they exist, in a proficient manner.…
Generalized Sliced Inverse Regression (GSIR) is one of the most important methods for nonlinear sufficient dimension reduction. As shown in Li and Song (2017), it enjoys a convergence rate that is independent of the dimension of the…
In this paper, we study regression problems over a separable Hilbert space with the square loss, covering non-parametric regression over a reproducing kernel Hilbert space. We investigate a class of spectral/regularized algorithms,…
The choice of parameterization in Nonlinear (NL) system models greatly affects the quality of the estimated model. Overly complex models can be impractical and hard to interpret, necessitating data-driven methods for simpler and more…
The curse of dimensionality presents a pervasive challenge in optimization problems, with exponential expansion of the search space rapidly causing traditional algorithms to become inefficient or infeasible. An adaptive sampling strategy is…
Kernel dimensionality reduction (KDR) algorithms find a low dimensional representation of the original data by optimizing kernel dependency measures that are capable of capturing nonlinear relationships. The standard strategy is to first…
Dimensionality reduction methods are unsupervised approaches which learn low-dimensional spaces where some properties of the initial space, typically the notion of "neighborhood", are preserved. Such methods usually require propagation on…
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.…
We present a collection of algorithms which utilize dimensional reduction to perform mesh refinement and study possibly singular solutions of time-dependent partial differential equations. The algorithms are inspired by constructions used…
We investigate nonparametric estimation of sliced inverse regression (SIR) via the $k$-nearest neighbors approach with a kernel. An estimator of the covariance matrix of the conditional expectation of the explanatory random vector given the…
We propose estimators based on kernel ridge regression for nonparametric causal functions such as dose, heterogeneous, and incremental response curves. Treatment and covariates may be discrete or continuous in general spaces. Due to a…
Dimensionality reduction (DR) methods have attracted extensive attention to provide discriminative information and reduce the computational burden of the hyperspectral image (HSI) classification. However, the DR methods face many challenges…
We propose a new data-driven approach for learning the fundamental solutions (Green's functions) of various linear partial differential equations (PDEs) given sample pairs of input-output functions. Building off the theory of functional…
This paper presents a comprehensive study on the convergence rates of the stochastic gradient descent (SGD) algorithm when applied to overparameterized two-layer neural networks. Our approach combines the Neural Tangent Kernel (NTK)…
We generalize Jan Willems' behavioral approach to a class of discrete-time nonlinear systems in a vector-valued reproducing kernel Hilbert space (RKHS). Apart from linear time-invariant systems, this class covers nonlinear systems modeled…
We propose a new randomized algorithm for solving L2-regularized least-squares problems based on sketching. We consider two of the most popular random embeddings, namely, Gaussian embeddings and the Subsampled Randomized Hadamard Transform…