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 dimensionality, as well as to the presence of noise, the classical approximation methods applicable in low dimensions are less effective in the high-dimensional case. We propose a new approximation method that leverages the advantages of the Manifold Locally Optimal Projection (MLOP) method (introduced by Faigenbaum-Golovin and Levin in 2020) and the strengths of the method of Radial Basis Functions (RBF). The method is parametrization free, requires no knowledge regarding the manifold's intrinsic dimension, can handle noise and outliers in both the function values and in the location of the data, and is applied directly in the high dimensions. We show that the complexity of the method is linear in the dimension of the manifold and squared-logarithmic in the dimension of the codomain of the function. Subsequently, we demonstrate the effectiveness of our approach by considering different manifold topologies and show the robustness of the method to various noise levels.
We consider a problem of great practical interest: the repairing and recovery of a low-dimensional manifold embedded in high-dimensional space from noisy scattered data. Suppose that we observe a point cloud sampled from the low-dimensional manifold, with noise, and let us assume that there are holes in the data. Can we recover missing information inside the holes? While in low-dimension the problem was extensively studied, manifold repairing in high dimension is still an open problem. We introduce a new approach, called Repairing Manifold Locally Optimal Projection (R-MLOP), that expands the …
In this paper, we present a method for denoising and reconstruction of low-dimensional manifold in high-dimensional space. We suggest a multidimensional extension of the Locally Optimal Projection algorithm which was introduced by Lipman et al. in 2007 for surface reconstruction in 3D. The method bypasses the curse of dimensionality and avoids the need for carrying out dimensional reduction. It is based on a non-convex optimization problem, which leverages a generalization of the outlier robust L1-median to higher dimensions while generating noise-free quasi-uniformly distributed points recons…
We compare a recently proposed multivariate spline based on mixed partial derivatives with two other standard splines for the scattered data smoothing problem. The splines are defined as the minimiser of a penalised least squares functional. The penalties are based on partial differentiation operators, and are integrated using the finite element method. We compare three methods to two problems: to remove the mixture of Gaussian and impulsive noise from an image, and to recover a continuous function from a set of noisy observations.
In this contribution, we introduce a multilevel approximation method with T-splines for fitting scattered point clouds iteratively, with an application to land remote sensing. This new procedure provides a local surface approximation by an explicit computation of the control points and is called a multilevel T-splines approximation (MTA). It is computationally efficient compared with the traditional global least-squares (LS) approach, which may fail when there is an unfavourable point density from a given refinement level. We validate our approach within a simulated framework and apply it to t…
Transformers for Learning on Noisy and Task-Level Manifolds: Approximation and Generalization Insights
School of Mathematics, Georgia Institute of Technology（佐治亚理工学院数学系）；Department of Mathematics, Purdue University（普渡大学数学系）；Department of Mathematics and Halicioğlu Data Science Institute, University of California, San Diego（加州大学圣地亚哥分校数学系和Halicioğlu数据科学研究所）
The eigenfunctions of the Laplace-Beltrami operator have widespread applications in a number of disciplines of engineering, computer vision/graphics, machine learning, etc. These eigenfunctions or manifold harmonics, provide the means to smoothly interpolate data on a manifold. They are highly effective, specifically as it relates to geometry representation and editing; manifold harmonics form a natural basis for multi-resolution representation (and editing) of complex surfaces and functioned defined therein. In this paper, we seek to develop the framework to exploit the benefits of manifold h…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学研究所）；University of New South Wales（新南威尔士大学）
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 problem of learning real-valued functions on manifolds through filtered hyperinterpolation of input-output data pairs where the inputs may be sampled deterministically or at random and the outputs may be clean or noisy. Motivated by the problem of handling large data sets, it presents a parallel data processing approach which distributes the data-fitting task…
