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Bayesian Interpolation [chapter]

David J. C. MacKay
1992 Maximum Entropy and Bayesian Methods  
Alternative regularizers (priors) and alternative basis sets are objectively compared by evaluating the evidence for them. "Occam's razor" is automatically embodied by this process.  ...  In this paper, the Bayesian approach to regularization and model-comparison is demonstrated by studying the inference problem of interpolating noisy data.  ...  For a radial basis function or "kernel" model, the basis functions are 4rl(x) = g[(xxh)/r]/r; here the xh are equally spaced over the range of interest.  ... 
doi:10.1007/978-94-017-2219-3_3 fatcat:ojegixvnjbhr7gamd46v2vish4

Bayesian Interpolation

David J. C. MacKay
1992 Neural Computation  
Alternative regularizers (priors) and alternative basis sets are objectively compared by evaluating the evidence for them. "Occam's razor" is automatically embodied by this process.  ...  In this paper, the Bayesian approach to regularization and model-comparison is demonstrated by studying the inference problem of interpolating noisy data.  ...  For a radial basis function or "kernel" model, the basis functions are 4rl(x) = g[(xxh)/r]/r; here the xh are equally spaced over the range of interest.  ... 
doi:10.1162/neco.1992.4.3.415 fatcat:gl2cbwdpx5dsdd67d7job3sxcu

Universal optimality of the E_8 and Leech lattices and interpolation formulas [article]

Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska
2022 arXiv   pre-print
It reconstructs a radial Schwartz function f from the values and radial derivatives of f and its Fourier transform f at the radii √(2n) for integers n≥1 in ℝ^8 and n ≥ 2 in ℝ^24.  ...  To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest.  ...  We also thank Princeton and the Institute for Advanced Study for hosting Viazovska during part of this work.  ... 
arXiv:1902.05438v3 fatcat:2oqv5tahbvdv7khz3zml6whlym

The Interpolation Theory of Radial Basis Functions [article]

Brad Baxter
2010 arXiv   pre-print
In this dissertation, it is first shown that, when the radial basis function is a p-norm and 1 < p < 2, interpolation is always possible when the points are all different and there are at least two of  ...  The greater part of this work investigates the sensitivity of radial basis function interpolants to changes in the function values at the interpolation points.  ...  Thus χ is the cardinal function of interpolation for the Gaussian radial basis function. Proposition 5.5.1.  ... 
arXiv:1006.2443v1 fatcat:zfu3m6qoirednpx5vepfvpve5i

Displacement interpolation using Lagrangian mass transport

Nicolas Bonneel, Michiel van de Panne, Sylvain Paris, Wolfgang Heidrich
2011 Proceedings of the 2011 SIGGRAPH Asia Conference on - SA '11  
Our method decomposes distributions or functions into sums of radial basis functions (RBFs).  ...  This paper develops the use of displacement interpolation for this class of problem, which provides a generic method for interpolating between distributions or functions based on advection instead of blending  ...  We acknowledge Stelian Coros for help with the character animation example, and reviewers for their detailed comments.  ... 
doi:10.1145/2024156.2024192 fatcat:vucyhbadmrelfduzzbpqfp4qla

Displacement interpolation using Lagrangian mass transport

Nicolas Bonneel, Michiel van de Panne, Sylvain Paris, Wolfgang Heidrich
2011 ACM Transactions on Graphics  
Our method decomposes distributions or functions into sums of radial basis functions (RBFs).  ...  This paper develops the use of displacement interpolation for this class of problem, which provides a generic method for interpolating between distributions or functions based on advection instead of blending  ...  We acknowledge Stelian Coros for help with the character animation example, and reviewers for their detailed comments.  ... 
doi:10.1145/2070781.2024192 fatcat:uzfdqizwlbfqvlrobmhe24wnmm

Displacement interpolation using Lagrangian mass transport

Nicolas Bonneel, Michiel van de Panne, Sylvain Paris, Wolfgang Heidrich
2011 Proceedings of the 2011 SIGGRAPH Asia Conference on - SA '11  
Our method decomposes distributions or functions into sums of radial basis functions (RBFs).  ...  This paper develops the use of displacement interpolation for this class of problem, which provides a generic method for interpolating between distributions or functions based on advection instead of blending  ...  We acknowledge Stelian Coros for help with the character animation example, and reviewers for their detailed comments.  ... 
doi:10.1145/2070752.2024192 fatcat:xv7avoqf5jehrnd46ghhdgvway

Kernel interpolation in Sobolev spaces is not consistent in low dimensions

Simon Buchholz
2022 Annual Conference Computational Learning Theory  
We show for d/2 < s < 3d/4 that interpolation is not consistent in fixed dimension, extending earlier results for the Laplace kernel in odd dimensions and underlining again that benign overfitting is rare  ...  The proof proceeds by deriving sharp bounds on the spectrum of random kernel matrices using results from the theory of radial basis functions which might be of independent interest.  ...  The radial basis function part We now state a lower bound on the spectrum of kernel matrices from the theory of radial basis function.  ... 
dblp:conf/colt/Buchholz22 fatcat:wzhtnzjb5vfjzhhkadb4kyaldm

PetRBF — A parallel O(N) algorithm for radial basis function interpolation with Gaussians

Rio Yokota, L.A. Barba, Matthew G. Knepley
2010 Computer Methods in Applied Mechanics and Engineering  
We have developed a parallel algorithm for radial basis function (RBF) interpolation that exhibits O(N) complexity,requires O(N) storage, and scales excellently up to a thousand processes.  ...  Previous fast RBF methods, --,achieving at most O(NlogN) complexity,--, were developed using multiquadric and polyharmonic basis functions.  ...  Acknowledgments Computing time provided by the Center for Computational Science (CCS) of Boston University, on Blue-Gene/L, and the UK National Supercomputing Service, on the hector supercomputer.  ... 
doi:10.1016/j.cma.2010.02.008 fatcat:teeh25zqa5bc7f3boe34vhjpze

A numerical framework for elastic surface matching, comparison, and interpolation [article]

Martin Bauer, Nicolas Charon, Philipp Harms, Hsi-Wei Hsieh
2020 arXiv   pre-print
By avoiding altogether the need for reparametrizations, it provides the flexibility to deal with simplicial meshes of arbitrary topologies and sampling patterns.  ...  Surface comparison and matching is a challenging problem in computer vision.  ...  kernel is of the form k(x 1 , n 1 , x 2 , n 2 ) := ρ(|x 1 − x 2 |)γ(n 1 · n 2 ) , (7) where ρ and γ are two functions defining a radial kernel on R 3 and a zonal kernel on S 2 , respectively.  ... 
arXiv:2006.11652v1 fatcat:3zoehcz66fchjlqebukwwpax34

Meshless Interpolations for Computer Graphics, Visualization and Games [article]

Vaclav Skala
2015 Eurographics State of the Art Reports  
Meshless Interpolations for Computer Graphics, Visualization and Games: An Informal Introduction  ...  Meshless Interpolations Eurographics, Zurich, 2015 Vaclav Skala  ...  Acknowledgment -some items included in this presentation were downloaded from the Internet open resources and authors are acknowledged if they are known. Thanks belong to them.  ... 
doi:10.2312/egt.20151046 fatcat:ogy6fmkqhnd37ngzuou2wxxgcq

Error estimates for interpolation of rough data using the scattered shifts of a radial basis function

R. A. Brownlee
2005 Numerical Algorithms  
The error between appropriately smooth functions and their radial basis function interpolants, as the interpolation points fill out a bounded domain in R^d, is a well studied artifact.  ...  In all of these cases, the analysis takes place in a natural function space dictated by the choice of radial basis function -- the native space.  ...  Now if Sf is the radial basis function interpolant to f then because f σ interpolates f , they can write Sf = Sf σ .  ... 
doi:10.1007/s11075-004-3620-2 fatcat:3g7op7f3ivf2jp75l63qjroqry

Interpolation Accuracy of Hybrid Soft Computing Techniques in Estimating Discharge Capacity of Triangular Labyrinth Weir

Ali Mahmoud, Xiaohui Yuan, Marwan Kheimi, Yanbin Yuan
2021 IEEE Access  
The hidden layer comprises several neurons that utilize a monotone non-increasing function as the activation function, namely, radial basis function.  ...  a: RADIAL BASIS FUNCTION NEURAL NETWORK (RBFNN) Broomhead and lowe [29] introduced RBFNN, which has been used extensively to address regression problems [30] .  ...  His research interests include mineral resource prospecting and exploration, geographical information systems, hydrology, and water resources. He is a member of IAMG and GSC.  ... 
doi:10.1109/access.2021.3049223 fatcat:xjan3g6b2bhb3ndhghmqhhlb44

Interpolation by periods in planar domain [article]

Mikhail Dubashinskiy
2015 arXiv   pre-print
For which Ω the collection of such period sequences coincides with ℓ^2? We give the answer in terms of metric properties of holes in Ω.  ...  The functional Per j has the unique continuous extension from closed and smooth in Ω L 2 -forms to the whole L 2,1 c (Ω).  ...  The multipliers technique used for investigation of interpolation problems in spaces of analytic functions seems not to be applicable in the case of periods as well as Blaschke products technique.  ... 
arXiv:1511.07015v1 fatcat:koorfaswknbojjmupzax2couka

Interpolation between Banach spaces and continuity of Radon-like integral transforms [article]

Pavel Zorin-Kranich
2013 arXiv   pre-print
The main new result concerns interpolation between H^1 and L^p estimates for analytic families of operators acting on Schwartz functions.  ...  We present the abstract framework and some applications of interpolation theory.  ...  n−3 2 rdr (r 2 = ρ 2 + t 2 ) Let s > R, multiply this equality by t(t 2 − s 2 ) (n−3)/2 and integrate for t ∈ (s, ∞).  ... 
arXiv:1301.1025v1 fatcat:i24pkpvtz5hcjdq2l3dgnkwbgu
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