Related papers: Approximating the Derivative of Manifold-valued Fu…
We propose a simple methodology to approximate functions with given asymptotic behavior by specifically constructed terms and an unconstrained deep neural network (DNN). The methodology we describe extends to various asymptotic behaviors…
We study in this paper the function approximation error of multivariate linear extrapolation. The sharp error bound of linear interpolation already exists in the literature. However, linear extrapolation is used far more often in…
In this paper I consider the applications of several kinds of approximations of real functions to the problem of verified computation (reliable computing) of the range of implicitly defined real function $x_{n+1} = G(x_{1}, ..., x_{n}),$…
In this short note, we propose an unified method to derive formulas for derivations conjugated by exponential functions on an almost complex manifold. In v3, we corrected some mistakes in previous versions.
Multivariate functions encountered in high-dimensional uncertainty quantification problems often vary most strongly along a few dominant directions in the input parameter space. We propose a gradient-based method for detecting these…
The objective of this paper is to design an embedding method that maps local features describing an image (e.g. SIFT) to a higher dimensional representation useful for the image retrieval problem. First, motivated by the relationship…
We investigate smooth approximations of functions, with prescribed gradient behavior on a distinguished stratified subset of the domain. As an application, we outline how our results yield important consequences for a recently introduced…
Multivariate function approximation is a fundamental problem in machine learning. Classic multivariate function approximations rely on hand-crafted basis functions (e.g., polynomial basis and Fourier basis), which limits their approximation…
We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in a Euclidean embedding space, and then projecting pointwise…
Manifold learning seeks a low dimensional representation that faithfully captures the essence of data. Current methods can successfully learn such representations, but do not provide a meaningful set of operations that are associated with…
We obtain approximation results for general positive linear operators satisfying mild conditions, when acting on discontinuous functions and absolutely continuous functions having discontinuous derivatives. The upper bounds, given in terms…
We study the computation of local approximations of invariant manifolds of parabolic fixed points and parabolic periodic orbits of periodic vector fields. If the dimension of these manifolds is two or greater, in general, it is not possible…
This paper studies the approximation of generalized ridge functions, namely of functions which are constant along some submanifolds of $\mathbb{R}^N$. We introduce the notion of linear-sleeve functions, whose function values only depend on…
Analyzing high-dimensional data with manifold learning algorithms often requires searching for the nearest neighbors of all observations. This presents a computational bottleneck in statistical manifold learning when observations of…
This paper considers the problem of embedding directed graphs in Euclidean space while retaining directional information. We model a directed graph as a finite set of observations from a diffusion on a manifold endowed with a vector field.…
The error term in the approximate functional equation for exponential sums involving the divisor function will be improved under certain conditions for the parameters of the approximate functional equation.
The concept of generalized functions taking values in a differentiable manifold is extended to a functorial theory. We establish several characterization results which allow a global intrinsic formulation both of the theory of…
Learning mappings of data on manifolds is an important topic in contemporary machine learning, with applications in astrophysics, geophysics, statistical physics, medical diagnosis, biochemistry, 3D object analysis. This paper studies the…
In this paper, we consider the fundamental problem of approximation of functions on a low-dimensional manifold embedded in a high-dimensional space, with noise affecting both in the data and values of the functions. Due to the curse of…
On any metric space, I provide an intrinsic characterization of those complex-valued functions which are uniform limits of Lipschitz functions. There are applications to function theory on complete Riemannian manifolds and, in particular,…