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Register a new userApproximation of Functions over Manifolds: A Moving Least-Squares Approach
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We present an algorithm for approximating a function defined over a $d$-dimensional manifold utilizing only noisy function values at locations sampled from the manifold with noise. To produce the approximation we do not require any knowledge regarding the manifold other than its dimension $d$. We use the Manifold Moving Least-Squares approach of (Sober and Levin 2016) to reconstruct the atlas of charts and the approximation is built on-top of those charts. The resulting approximant is shown to be a function defined over a neighborhood of a manifold, approximating the originally sampled manifold. In other words, given a new point, located near the manifold, the approximation can be evaluated directly on that point. We prove that our construction yields a smooth function, and in case of noiseless samples the approximation order is $mathcal{O}(h^{m+1})$, where $h$ is a local density of sample parameter (i.e., the fill distance) and $m$ is the degree of a local polynomial approximation, used in our algorithm. In addition, the proposed algorithm has linear time complexity with respect to the ambient-spaces dimension. Thus, we are able to avoid the computational complexity, commonly encountered in high dimensional approximations, without having to perform non-linear dimension reduction, which inevitably introduces distortions to the geometry of the data. Additionaly, we show numerical experiments that the proposed approach compares favorably to statistical approaches for regression over manifolds and show its potential.
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In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a lower dimensional manifold, thus the high dimensionality problem can be overcome by learning the lower dimensionality behavior. However, in real life applications, data is often very noisy. In this work, we propose a method to approximate $mathcal{M}$ a $d$-dimensional $C^{m+1}$ smooth submanifold of $mathbb{R}^n$ ($d ll n$) based upon noisy scattered data points (i.e., a data cloud). We assume that the data points are located near the lower dimensional manifold and suggest a non-linear moving least-squares projection on an approximating $d$-dimensional manifold. Under some mild assumptions, the resulting approximant is shown to be infinitely smooth and of high approximation order (i.e., $O(h^{m+1})$, where $h$ is the fill distance and $m$ is the degree of the local polynomial approximation). The method presented here assumes no analytic knowledge of the approximated manifold and the approximation algorithm is linear in the large dimension $n$. Furthermore, the approximating manifold can serve as a framework to perform operations directly on the high dimensional data in a computationally efficient manner. This way, the preparatory step of dimension reduction, which induces distortions to the data, can be avoided altogether.
Edge bundling methods can effectively alleviate visual clutter and reveal high-level graph structures in large graph visualization. Researchers have devoted significant efforts to improve edge bundling according to different metrics. As the edge bundling family evolve rapidly, the quality of edge bundles receives increasing attention in the literature accordingly. In this paper, we present MLSEB, a novel method to generate edge bundles based on moving least squares (MLS) approximation. In comparison with previous edge bundling methods, we argue that our MLSEB approach can generate better results based on a quantitative metric of quality, and also ensure scalability and the efficiency for visualizing large graphs.
Point set is a flexible and lightweight representation widely used for 3D deep learning. However, their discrete nature prevents them from representing continuous and fine geometry, posing a major issue for learning-based shape generation. In this work, we turn the discrete point sets into smooth surfaces by introducing the well-known implicit moving least-squares (IMLS) surface formulation, which naturally defines locally implicit functions on point sets. We incorporate IMLS surface generation into deep neural networks for inheriting both the flexibility of point sets and the high quality of implicit surfaces. Our IMLSNet predicts an octree structure as a scaffold for generating MLS points where needed and characterizes shape geometry with learned local priors. Furthermore, our implicit function evaluation is independent of the neural network once the MLS points are predicted, thus enabling fast runtime evaluation. Our experiments on 3D object reconstruction demonstrate that IMLSNets outperform state-of-the-art learning-based methods in terms of reconstruction quality and computational efficiency. Extensive ablation tests also validate our network design and loss functions.
The approximation of both geodesic distances and shortest paths on point cloud sampled from an embedded submanifold $mathcal{M}$ of Euclidean space has been a long-standing challenge in computational geometry. Given a sampling resolution parameter $ h $, state-of-the-art discrete methods yield $ O(h) $ provable approximations. In this paper, we investigate the convergence of such approximations made by Manifold Moving Least-Squares (Manifold-MLS), a method that constructs an approximating manifold $mathcal{M}^h$ using information from a given point cloud that was developed by Sober & Levin in 2019. In this paper, we show that provided that $mathcal{M}in C^{k}$ and closed (i.e. $mathcal{M}$ is a compact manifold without boundary) the Riemannian metric of $ mathcal{M}^h $ approximates the Riemannian metric of $ mathcal{M}, $. Explicitly, given points $ p_1, p_2 in mathcal{M}$ with geodesic distance $ rho_{mathcal{M}}(p_1, p_2) $, we show that their corresponding points $ p_1^h, p_2^h in mathcal{M}^h$ have a geodesic distance of $ rho_{mathcal{M}^h}(p_1^h,p_2^h) = rho_{mathcal{M}}(p_1, p_2)(1 + O(h^{k-1})) $ (i.e., the Manifold-MLS is nearly an isometry). We then use this result, as well as the fact that $ mathcal{M}^h $ can be sampled with any desired resolution, to devise a naive algorithm that yields approximate geodesic distances with a rate of convergence $ O(h^{k-1}) $. We show the potential and the robustness to noise of the proposed method on some numerical simulations.
We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.
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