Related papers: A least squares approach to Whitney forms
The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge…
Inspired by recent developments in subdivision schemes founded on the Weighted Least Squares technique, we construct linear approximants for noisy data in which the weighting strategy minimizes the output variance, thereby establishing a…
We propose a new least squares finite element method to solve the Stokes problem with two sequential steps. The approximation spaces are constructed by patch reconstruction with one unknown per element. For the first step, we reconstruct an…
The Whitney embedding theorem gives an upper bound on the smallest embedding dimension of a manifold. If a data set lies on a manifold, a random projection into this reduced dimension will retain the manifold structure. Here we present an…
We describe stochastic Newton and stochastic quasi-Newton approaches to efficiently solve large linear least-squares problems where the very large data sets present a significant computational burden (e.g., the size may exceed computer…
Data approximation is essential in fields such as geometric design, numerical PDEs, and curve modeling. Moving Least Squares (MLS) is a widely used method for data fitting; however, its accuracy degrades in the presence of discontinuities,…
Recursive estimates of large systems of equations in the context of least squares fitting is a common practice in different fields of study. For example, recursive adaptive filtering is extensively used in signal processing and control…
We study simplicial complexes with a given number of vertices whose Stanley-Reisner ring has the minimal possible Betti numbers. We find that these simplicial complexes have very special combinatorial and topological structures. For…
We establish a set of relations between several quite diverse types of weighted inequalities involving various integral operators and fairly general quasinorm-like functionals which we call sub-monotone. The main result enables one to solve…
We propose a new least squares finite element method to solve the Poisson equation. By using a piecewisely irrotational space to approximate the flux, we split the classical method into two sequential steps. The first step gives the…
The function-on-function linear regression model in which the response and predictors consist of random curves has become a general framework to investigate the relationship between the functional response and functional predictors.…
We discuss the abstract structure of sequential weak measurement (WM) of general observables. In all orders, the sequential WM correlations without post-selection yield the corresponding correlations of the Wigner function, offering direct…
In this article, we present weighted norm inequality for a fractional one-sided minimal function. We prove weighted weak and strong type norm inequalities for the one-sided minimal function on $\mathbb{R}.$ We construct two weight classes…
We develop the Lifshitz theory of van der Waals forces in a wedge of a dielectric material. The non-planar geometry of the problem requires determining point-wise distribution of stresses. The findings are relevant to a wide range of…
Traditionally, the least squares regression is mainly concerned with studying the effects of individual predictor variables, but strongly correlated variables generate multicollinearity which makes it difficult to study their effects.…
We consider the problem of nonparametric regression under shape constraints. The main examples include isotonic regression (with respect to any partial order), unimodal/convex regression, additive shape-restricted regression, and…
We propose an abstract framework for analyzing the convergence of least-squares methods based on residual minimization when feasible solutions are neural networks. With the norm relations and compactness arguments, we derive error estimates…
In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also…
This paper investigates the behavior of a heavy soft spring in steady circular motion. Since the spring is inhomogeneous due to centrifugal force, one can rigorously prove that it follows the one-dimensional static Willis-form equations.…
We develop a technique for establishing lower bounds on the sample complexity of Least Squares (or, Empirical Risk Minimization) for large classes of functions. As an application, we settle an open problem regarding optimality of Least…