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We study the expressibility and learnability of convex optimization solution functions and their multi-layer architectural extension. The main results are: \emph{(1)} the class of solution functions of linear programming (LP) and quadratic…
This paper presents a description and analysis of a rational cubic spline FIF (RCSFIF) that has two shape parameters in each subinterval when it is defined implicitly. To be precise, we consider the iterated function system (IFS) with…
This article characterizes the exact asymptotics of random Fourier feature (RFF) regression, in the realistic setting where the number of data samples $n$, their dimension $p$, and the dimension of feature space $N$ are all large and…
An aperiodic (low frequency) spectrum may originate from the error term in the mean value of an arithmetical function such as M\"obius function or Mangoldt function, which are coding sequences for prime numbers. In the discrete Fourier…
The quadrature error associated with a regular quadrature rule for evaluation of a layer potential increases rapidly when the evaluation point approaches the surface and the integral becomes nearly singular. Error estimates are needed to…
Three kinds of effective error bounds of the quadrature formulas with multiple nodes that are generalizations of the well known Micchelli-Rivlin quadrature formula, when the integrand is a function analytic in the regions bounded by…
Scattered data interpolation schemes using kriging and radial basis functions (RBFs) have the advantage of being meshless and dimensional independent, however, for the data sets having insufficient observations, RBFs have the advantage over…
We provide tools to help automate the error analysis of algorithms that evaluate simple functions over the floating-point numbers. The aim is to obtain tight relative error bounds for these algorithms, expressed as a function of the unit…
Accurate material modeling is crucial for achieving photorealistic rendering, bridging the gap between computer-generated imagery and real-world photographs. While traditional approaches rely on tabulated BRDF data, recent work has shifted…
Kernel methods give powerful, flexible, and theoretically grounded approaches to solving many problems in machine learning. The standard approach, however, requires pairwise evaluations of a kernel function, which can lead to scalability…
Approximation of scattered data is often a task in many engineering problems. The Radial Basis Function (RBF) approximation is appropriate for big scattered datasets in $n-$dimensional space. It is a non-separable approximation, as it is…
The direct method used for calculating smooth radial basis function (RBF) interpolants in the flat limit becomes numerically unstable. The RBF-QR algorithm bypasses this ill-conditioning using a clever change of basis technique. We extend…
Fourier embedding has shown great promise in removing spectral bias during neural network training. However, it can still suffer from high generalization errors, especially when the labels or measurements are noisy. We demonstrate that…
Most problems in electrodynamics do not have an analytical solution so much effort has been put in the development of numerical schemes, such as the finite-difference method, volume element methods, boundary element methods, and related…
The reconstruction fidelity of computational optical imaging is fundamentally constrained by the model-reality gap, i.e., the inevitable discrepancy between idealized forward models and the physical imaging process. Conventional paradigms…
Operator learning is a data-driven approximation of mappings between infinite-dimensional function spaces, such as the solution operators of partial differential equations. Kernel-based operator learning can offer accurate, theoretically…
Statistical inferences for quadratic functionals of linear regression parameter have found wide applications including signal detection, global testing, inferences of error variance and fraction of variance explained. Classical theory based…
One of the distinguishing characteristics of modern deep learning systems is that they typically employ neural network architectures that utilize enormous numbers of parameters, often in the millions and sometimes even in the billions.…
Nonlinear dimensionality reduction embeddings computed from datasets do not provide a mechanism to compute the inverse map. In this paper, we address the problem of computing a stable inverse map to such a general bi-Lipschitz map. Our…
This paper introduces a geometric theory of model error, treating true and model dynamics as geodesic flows generated by distinct affine connections on a smooth manifold. When these connections differ, the resulting trajectory…