Quadrature rules and distribution of points on manifolds

Brandolini, Luca; Choirat, Christine; Colzani, Leonardo; Gigante, Giacomo; Seri, Raffaello; Travaglini, Giancarlo

doi:10.2422/2036-2145.201103_007

Cited by 40 publications

(82 citation statements)

References 30 publications

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“…The sampling point set of the data set D satisfies Assumption 8. Then, for c 3 6 Here, in contrast to Theorem 6, y contains noise. The expectation in ( 14) is with respect to the noise on y.…”

Section: Non-distributed Filtered Hyperinterpolation For Noisy Datamentioning

confidence: 93%

“…For general Riemannian manifolds, the construction of quadrature formulas depends on the existence of designs on manifolds, which has been proved in [15]. See also [3]. On a flat torus, the weights are equal for a regular grid in [−π, π] d .…”

Section: Non-distributed Filtered Hyperinterpolation On Manifoldsmentioning

confidence: 99%

“…These give M) and V D * j ,n − f * 2 L 2 (M) . By min j=1,...,m |D j | ≥ |D| 2d/(2r +d) and D j and Assumption 8, there exists a quadrature rule for each local server which is exact for polynomials of degree 3n − 1 for n satisfying c 3 6 |D| 1/(2r +d) ≤ n ≤ For each j = 1, . .…”

Section: Lemma 16mentioning

confidence: 99%

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Distributed Learning via Filtered Hyperinterpolation on Manifolds

Montúfar

Wang

2021

Found Comput Math

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Learning mappings of data on manifolds is an important topic in contemporary machine learning, with applications in astrophysics, geophysics, statistical physics, medical diagnosis, biochemistry, and 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 among multiple servers and synthesizes the fitted sub-models into a global estimator. We prove quantitative relations between the approximation quality of the learned function over the entire manifold, the type of target function, the number of servers, and the number and type of available samples. We obtain the approximation rates of convergence for distributed and non-distributed approaches. For the non-distributed case, the approximation order is optimal.

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Section: Non-distributed Filtered Hyperinterpolation For Noisy Datamentioning

confidence: 93%

Section: Non-distributed Filtered Hyperinterpolation On Manifoldsmentioning

confidence: 99%

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Distributed Learning via Filtered Hyperinterpolation on Manifolds

Montúfar

Wang

2021

Found Comput Math

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“…3; deterministic quadrature schemes provide a more uniform sampling of the integral domain compared to pseudorandomly sampled quadrature points, which naturally appear in clusters or groupings. The uniformity of sampling is quantified by the discrepancy of the samples [24][25][26][27][28], with low discrepancy samples providing the most uniform coverage. (Recent attention has focused on deploying quasi-Monte Carlo schemes to solve radiative transfer problems, which circumvent the issue of high discrepancy as described later in the paper.…”

Section: Traditional Applications Of Monte Carlo For Modeling Radiative Transfermentioning

confidence: 99%

The Past and Future of the Monte Carlo Method in Thermal Radiation Transfer

Howell

Daun

2021

Journal of Heat Transfer

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“…On general manifolds, some results are obtained in Mhaskar [18,Theorem 3.3], which relies on approximation by eigenfunctions of the Laplace-Beltrami operator on the manifold. In the general case, more is known about recovering only an integral instead of the function itself, see Brandolini et al [2]. The proofs behind many of these results rely on Marcinkiewicz-Zygmund inequalities and exact integration of such eigenfunctions by quadrature rules, again using evenly spaced quasi-uniform points, see for example Filbir and Mhaskar [10].…”

mentioning

confidence: 99%

Function recovery on manifolds using scattered data

Krieg¹,

Sonnleitner²

2021

Preprint

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We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold M when given a sample on a finite point set. We prove that the quality of the sample is given by the L γ (M )-average of the geodesic distance to the point set and determine the value of γ ∈ (0, ∞]. This extends our findings on bounded convex domains [arXiv:2009[arXiv: .11275, 2020. Further, a limit theorem for moments of the average distance to a set consisting of i.i.d. uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with γ < ∞. In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput., 29:1203-1214, 2019.Disclaimer. This paper relies on our previous work [15] which is still in the revision process. The reader may therefore not yet find a complete proof of all the results that are presented in this paper. However, this only concerns the lower bound of Theorem 2 for smoothness s ∈ N and the lower bound of Corollary 1 in the case that the dimension d is even. See Remark 2 below.We are interested in the problem of recovering a real-valued function on a connected compact Riemannian manifold which is only known through a finite number of function values, obtained on a finite set of sampling points. We study this problem

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Quadrature rules and distribution of points on manifolds

Cited by 40 publications

References 30 publications

Distributed Learning via Filtered Hyperinterpolation on Manifolds

Distributed Learning via Filtered Hyperinterpolation on Manifolds

The Past and Future of the Monte Carlo Method in Thermal Radiation Transfer

Function recovery on manifolds using scattered data

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