On the Error of Random Sampling: Uniformly Distributed Random Points on Parametric Curves
Pages 273 - 282
Abstract
Given a parametric polynomial curve γ:[a,b] →Rn, how can we sample a random point x ∈ im(γ) in such a way that it is distributed uniformly with respect to the arc-length? Unfortunately, we cannot sample exactly such a point---even assuming we can perform exact arithmetic operations. So we end up with the following question: how does the method we choose affect the quality of the approximate sample we obtain? In practice, there are many answers. However, in theory, there are still gaps in our understanding. In this paper, we address this question from the point of view of complexity theory, providing bounds in terms of the size of the desired error.
References
[1]
Lenore Blum, Felipe Cucker, Michael Shub, and Steve Smale. 1998. Complexity and real computation. Springer-Verlag, New York. xvi+453 pages. https://doi.org/10.1007/978--1--4612-0701--6 With a foreword by Richard M. Karp.
[2]
Paul Breiding, Sara Kali'snik, Bernd Sturmfels, and Madeleine Weinstein. 2018. Learning algebraic varieties from samples. Rev. Mat. Complut. 31, 3 (2018), 545--593. https://doi.org/10.1007/s13163-018-0273--6
[3]
Zongchen Chen and Santosh S. Vempala. 2019. Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions. In APPROX/ RANDOM 2019 (LIPIcs, Vol. 145), Dimitris Achlioptas and L´aszl´o A. V´egh (Eds.). Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, Wadern, Germany, 64:1--64:12. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.64
[4]
Siddhartha Chib and Edward Greenberg. 1995. Understanding the metropolishastings algorithm. The american statistician 49, 4 (1995), 327--335. https://doi.org/10.1080/00031305.1995.10476177
[5]
Raaz Dwivedi, Yuansi Chen, Martin JWainwright, and Bin Yu. 2019. Log-concave sampling: Metropolis-Hastings algorithms are fast. Journal of Machine Learning Research 20, 183 (2019), 1--42.
[6]
Luiz Henrique de Figueiredo. 1995. IV.4 - Adaptive Sampling of Parametric Curves. In Graphics Gems V, Alan W. Paeth (Ed.). Academic Press, Boston, 173--178. https://doi.org/10.1016/B978-0--12--543457--7.50032--2
[7]
Michael S. Floater and Atgeirr F. Rasmussen. 2006. Point-based methods for estimating the length of a parametric curve. J. Comput. Appl. Math. 196, 2 (2006), 512--522. https://doi.org/10.1016/j.cam.2005.10.001
[8]
Michael S. Floater, Atgeirr F. Rasmussen, and Ulrich Reif. 2007. Extrapolation methods for approximating arc length and surface area. Numer. Algorithms 44, 3 (2007), 235--248. https://doi.org/10.1007/s11075-007--9095--1
[9]
L. Fox and I. B. Parker. 1968. Chebyshev polynomials in numerical analysis. Oxford University Press, London-New York-Toronto, Ont. ix+205 pages.
[10]
W. R. Gilks and P. Wild. 1992. Adaptive Rejection Sampling for Gibbs Sampling. Journal of the Royal Statistical Society. Series C (Applied Statistics) 41, 2 (1992), 337--348. http://www.jstor.org/stable/2347565
[11]
Jens Gravesen. 1997. Adaptive subdivision and the length and energy of B´ezier curves. Comput. Geom. 8, 1 (1997), 13--31. https://doi.org/10.1016/0925--7721(95) 00054--2
[12]
James Johndrow and Aaron Smith. 2018. Fast mixing of metropolis-hastings with unimodal targets. Electronic Communications in Probability 23, none (2018), 1 -- 9. https://doi.org/10.1214/18-ECP170
[13]
Ravi Kannan and Santosh Vempala. 1997. Sampling Lattice Points., 5 pages. https://doi.org/10.1145/258533.258665
[14]
Yin Tat Lee, Zhao Song, and Santosh S Vempala. 2018. Algorithmic theory of ODEs and sampling from well-conditioned log-concave densities. arXiv:1812.06243.
[15]
Josef Leydold. 1998. A Rejection Technique for Sampling from Log-Concave Multivariate Distributions. ACM Trans. Model. Comput. Simul. 8, 3 (jul 1998), 254--280. https://doi.org/10.1145/290274.290287
[16]
Laszlo Lovasz and Santosh Vempala. 2006. Fast Algorithms for Logconcave Functions: Sampling, Rounding, Integration and Optimization. In 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06). IEEE, Berkeley, CA, USA, 57--68. https://doi.org/10.1109/FOCS.2006.28
[17]
Laszlo Lovasz and Santosh Vempala. 2006. Hit-and-run from a corner. SIAM J. Comput. 35, 4 (2006), 985--1005.
[18]
Oren Mangoubi and Nisheeth K Vishnoi. 2019. Nonconvex sampling with the Metropolis-adjusted Langevin algorithm. In Proceedings of the Thirty-Second Conference on Learning Theory (Proceedings of Machine Learning Research, Vol. 99), Alina Beygelzimer and Daniel Hsu (Eds.). PMLR, Phoenix, USA, 2259--2293. https://proceedings.mlr.press/v99/mangoubi19a.html
[19]
J. C. Mason and D. C. Handscomb. 2003. Chebyshev polynomials. Chapman & Hall/CRC, Boca Raton, FL. xiv+341 pages.
[20]
Radford M. Neal. 2003. Slice sampling. Ann. Statist. 31, 3 (06 2003), 705--767. https://doi.org/10.1214/aos/1056562461
[21]
Radford M. Neal. 2011. MCMC using Hamiltonian dynamics. In Handbook of Markov chain Monte Carlo. CRC Press, Boca Raton, FL, USA, 113--162.
[22]
Partha Niyogi, Stephen Smale, and Shmuel Weinberger. 2008. Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39, 1--3 (2008), 419--441. https://doi.org/10.1007/s00454-008--9053--2
[23]
Sheehan Olver and Alex Townsend. 2013. Fast inverse transform sampling in one and two dimensions. arXiv:1307.1223
[24]
Luca Pagani and Paul J. Scott. 2018. Curvature based sampling of curves and surfaces. Computer Aided Geometric Design 59 (2018), 32--48. https://doi.org/10.1016/j.cagd.2017.11.004
[25]
Les Piegl and Wayne Tiller. 1997. The NURBS Book (2nd Ed.). Springer-Verlag, Berlin, Heidelberg.
[26]
R.L. Smith. 1996. The hit-and-run sampler: a globally reaching markov chain sampler for generating arbitrary multivariate distributions. In Proceedings Winter Simulation Conference. IEEE, Coronado, CA, USA, 260--264. https://doi.org/10.1109/WSC.1996.873287
[27]
J. Tonelli-Cueto and E. Tsigaridas. 2021. Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces. To appear in the special issue of the Journal of Symbolic Computation for ISSAC 2020. Available at arXiv:2006.04423.
[28]
Lloyd N. Trefethen. 2013. Approximation theory and approximation practice. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. viii+305 pp.+back matter pages.
[29]
Santosh Vempala. 2005. Geometric random walks: a survey. Combinatorial and computational geometry 52, 573--612 (2005), 2.
[30]
Marcelo Walter and Alain Fournier. 1996. Approximate arc length parameterization. In Proceedings of the 9th Brazilian symposium on computer graphics and image processing. Citeseer, Caxambu, Minas Gerais, Brazil, 143--150.
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Published: 05 July 2022
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- Campbs France
- Agence nationale de la recherche
- Fondation Mathématique Jacques Hadamard
- Fondation Sciences Mathématiques de Paris
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ISSAC '22
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ISSAC '22: International Symposium on Symbolic and Algebraic Computation
July 4 - 7, 2022
Villeneuve-d'Ascq, France
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