Error Guarantees for Least Squares Approximation with Noisy Samples in Domain Adaptation

Felix Bartel

doi:10.5802/smai-jcm.96

Felix Bartel ¹

¹ Chemnitz University of Technology, Faculty of Mathematics, 09107 Chemnitz, Germany

The SMAI Journal of computational mathematics, Volume 9 (2023), pp. 95-120.

Abstract

Given $n$ samples of a function $f : D \to ℂ$ in random points drawn with respect to a measure $ϱ_{S}$ we develop theoretical analysis of the $L_{2} (D, ϱ_{T})$ -approximation error. For a parituclar choice of $ϱ_{S}$ depending on $ϱ_{T}$ , it is known that the weighted least squares method from finite dimensional function spaces $V_{m}$ , $dim (V_{m}) = m < \infty$ has the same error as the best approximation in $V_{m}$ up to a multiplicative constant when given exact samples with logarithmic oversampling. If the source measure $ϱ_{S}$ and the target measure $ϱ_{T}$ differ we are in the domain adaptation setting, a subfield of transfer learning. We model the resulting deterioration of the error in our bounds.

Further, for noisy samples, our bounds describe the bias-variance trade off depending on the dimension $m$ of the approximation space $V_{m}$ . All results hold with high probability.

For demonstration, we consider functions defined on the $d$ -dimensional cube given in unifom random samples. We analyze polynomials, the half-period cosine, and a bounded orthonormal basis of the non-periodic Sobolev space $H_{mix}^{2}$ . Overcoming numerical issues of this $H_{mix}^{2}$ basis, this gives a novel stable approximation method with quadratic error decay. Numerical experiments indicate the applicability of our results.

Published online: 2023-07-31

DOI: 10.5802/smai-jcm.96

Classification: 41A10, 41A25, 41A60, 41A63, 42C10, 65TXX, 65F22, 65D15, 94A20
Keywords: domain adaptation, individual function approximation, least squares, sampling theory, transfer learning, unit cube, polynomial approximation

Author's affiliations:

Felix Bartel ¹

¹ Chemnitz University of Technology, Faculty of Mathematics, 09107 Chemnitz, Germany

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Felix Bartel},
     title = {Error {Guarantees} for {Least} {Squares} {Approximation} with {Noisy} {Samples} in {Domain} {Adaptation}},
     journal = {The SMAI Journal of computational mathematics},
     pages = {95--120},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Felix Bartel. Error Guarantees for Least Squares Approximation with Noisy Samples in Domain Adaptation. The SMAI Journal of computational mathematics, Volume 9 (2023), pp. 95-120. doi : 10.5802/smai-jcm.96. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.96/

References
Cited by

[1] B. Adcock Multivariate modified Fourier series and application to boundary value problems, Numer. Math., Volume 115 (2010) no. 4, pp. 511-552 | DOI | MR | Zbl

[2] B. Adcock; D. Huybrechs Multivariate modified Fourier expansions, Spectral and high order methods for partial differential equations (Lecture Notes in Computational Science and Engineering), Volume 76, Springer, 2011, pp. 85-92 | DOI | MR | Zbl

[3] B. Adcock; A. Iserles; S. P. Nørsett From high oscillation to rapid approximation II: expansions in Birkhoff series, IMA J. Numer. Anal., Volume 32 (2012) no. 1, pp. 105-140 | DOI | MR | Zbl

[4] Y. Baraud Model selection for regression on a random design, ESAIM, Probab. Stat., Volume 6 (2002), pp. 127-146 | DOI | Numdam | MR | Zbl

[5] F. Bartel; R. Hielscher Concentration inequalities for cross-validation in scattered data approximation, J. Approx. Theory, Volume 277 (2022), 105715, 17 pages | DOI | MR | Zbl

[6] F. Bartel; R. Hielscher; D. Potts Fast Cross-validation in Harmonic Approximation, Appl. Comput. Harmon. Anal., Volume 49 (2020) no. 2, pp. 415-437 | DOI | MR | Zbl

[7] F. Bartel; M. Schäfer; T. Ullrich Constructive subsampling of finite frames with applications in optimal function recovery, Appl. Comput. Harmon. Anal. (2023) (to appear) | DOI | MR | Zbl

[8] P. C. Bellec Concentration of quadratic forms under a Bernstein moment assumption (2019) (https://arxiv.org/abs/1901.08736)

[9] A. Chkifa; A. Cohen; G. Migliorati; F. Nobile; R. Tempone Discrete least squares polynomial approximation with random evaluations—application to parametric and stochastic elliptic PDEs, ESAIM, Math. Model. Numer. Anal., Volume 49 (2015) no. 3, pp. 815-837 | DOI | Numdam | MR | Zbl

[10] A. Cohen; M. A. Davenport; D. Leviatan On the stability and accuracy of least squares approximations, Found. Comput. Math., Volume 13 (2013) no. 5, pp. 819-834 | DOI | MR | Zbl

[11] A. Cohen; G. Migliorati Optimal weighted least-squares methods, SMAI J. Comput. Math., Volume 3 (2017), pp. 181-203 | DOI | Numdam | MR | Zbl

[12] R. Cools; F. Y. Kuo; D. Nuyens; G. Suryanarayana Tent-transformed lattice rules for integration and approximation of multivariate non-periodic functions, J. Complexity, Volume 36 (2016), pp. 166-181 | DOI | MR | Zbl

[13] J. Dick; D. Nuyens; F. Pillichshammer Lattice rules for nonperiodic smooth integrands, Numer. Math., Volume 126 (2014) no. 2, pp. 259-291 | DOI | MR | Zbl

[14] M. Dolbeault; A. Cohen Optimal pointwise sampling for $L^{2}$ approximation, J. Complexity, Volume 68 (2022), 101602 | DOI | MR | Zbl

[15] M. Dolbeault; A. Cohen Optimal sampling and Christoffel functions on general domains, Constr. Approx., Volume 56 (2022) no. 1, pp. 121-163 | DOI | MR | Zbl

[16] M. Dolbeault; D. Krieg; M. Ullrich A sharp upper bound for sampling numbers in $L_{2}$ , Appl. Comput. Harmon. Anal., Volume 63 (2023), pp. 113-134 | DOI | MR

[17] S. Foucart; H. Rauhut A mathematical introduction to compressive sensing, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, 2013, xviii+625 pages | DOI | MR

[18] E. R. Gizewski; L. Mayer; B. A. Moser; D. H. Nguyen; S. Pereverzyev; S. V. Pereverzyev; N. Shepeleva; W. Zellinger On a regularization of unsupervised domain adaptation in RKHS, Appl. Comput. Harmon. Anal., Volume 57 (2022), pp. 201-227 | DOI | MR | Zbl

[19] A. Greenbaum Iterative methods for solving linear systems, Frontiers in Applied Mathematics, 17, Society for Industrial and Applied Mathematics, 1997, xiv+220 pages | DOI | MR

[20] L. Györfi; M. Kohler; A. Krzyżak; H. Walk A distribution-free theory of nonparametric regression, Springer Series in Statistics, Springer, 2002, xvi+647 pages | DOI | MR

[21] C. Haberstich; A. Nouy; G. Perrin Boosted optimal weighted least-squares, Math. Comput. (2022) | DOI | MR | Zbl

[22] J. Hampton; A. Doostan Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression, Comput. Methods Appl. Mech. Eng., Volume 290 (2015), pp. 73-97 | DOI | MR | Zbl

[23] A. Iserles; S. P. Nørsett From high oscillation to rapid approximation. I. Modified Fourier expansions, IMA J. Numer. Anal., Volume 28 (2008) no. 4, pp. 862-887 | DOI | MR | Zbl

[24] L. Kämmerer; T. Ullrich; T. Volkmer Worst-case recovery guarantees for least squares approximation using random samples, Constr. Approx., Volume 54 (2021) no. 2, pp. 295-352 | DOI | MR | Zbl

[25] M. G. Krein On a special class of differential operators, Dokl. Akad. Nauk SSSR, Volume 2 (1935), pp. 345-349

[26] D. Krieg; M. Ullrich Function values are enough for $L_{2}$ -approximation, Found. Comput. Math., Volume 21 (2021) no. 4, pp. 1141-1151 | DOI | MR | Zbl

[27] F. Y. Kuo; G. Migliorati; F. Nobile; D. Nuyens Function integration, reconstruction and approximation using rank-1 lattices, Math. Comput., Volume 90 (2021) no. 330, pp. 1861-1897 | DOI | MR | Zbl

[28] I. Limonova; V. N. Temlyakov On sampling discretization in $L_{2}$ , J. Math. Anal. Appl., Volume 515 (2022) no. 2, 126457, 14 pages | DOI | MR | Zbl

[29] L. Lippert; D. Potts; T. Ullrich Fast Hyperbolic Wavelet Regression meets ANOVA (2021) (https://arxiv.org/abs/2108.13197, to appear in Numer. Math.) | DOI

[30] S. Lu; P. Mathé; S. V. Pereverzev Balancing principle in supervised learning for a general regularization scheme, Appl. Comput. Harmon. Anal., Volume 48 (2020) no. 1, pp. 123-148 | DOI | MR | Zbl

[31] G. Migliorati; F. Nobile; E. von Schwerin; R. Tempone Analysis of Discrete $L^{2}$ Projection on Polynomial Spaces with Random Evaluations, Found. Comput. Math. (2014) | DOI | MR | Zbl

[32] M. Moeller; T. Ullrich $L_{2}$ -norm sampling discretization and recovery of functions from RKHS with finite trace, Sampl. Theory Signal Process. Data Anal., Volume 19 (2021) no. 2, 13, 31 pages | DOI | MR | Zbl

[33] N. Nagel; M. Schäfer; T. Ullrich A New Upper Bound for Sampling Numbers, Found. Comput. Math. (2021) | DOI

[34] A. Narayan; J. D. Jakeman; T. Zhou A Christoffel function weighted least squares algorithm for collocation approximations, Math. Comput., Volume 86 (2017) no. 306, pp. 1913-1947 | DOI | MR | Zbl

[35] R. Nasdala; D. Potts A note on transformed Fourier systems for the approximation of non-periodic signals, Monte Carlo and quasi-Monte Carlo methods (Springer Proceedings in Mathematics & Statistics), Volume 387, Springer, 2022, pp. 253-271 | DOI | MR

[36] S. J. Pan; Q. Yang A Survey on Transfer Learning, IEEE Trans. Knowl. Data Eng., Volume 22 (2010) no. 10, pp. 1345-1359 | DOI

[37] S. V. Pereverzyev; S. Lu Regularization Theory for Ill-Posed Problems. Selected Topics, Walter de Gruyter, 2013, 287 pages | DOI

[38] G. Plonka; D. Potts; G. Steidl; M. Tasche Numerical Fourier analysis, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, 2018, xvi+168 pages | DOI | MR

[39] D. Potts; M. Schmischke Learning multivariate functions with low-dimensional structures using polynomial bases, J. Comput. Appl. Math., Volume 403 (2022), 113821, 19 pages | DOI | MR | Zbl

[40] D. Potts; T. Volkmer Fast and exact reconstruction of arbitrary multivariate algebraic polynomials in Chebyshev form, 2015 International Conference on Sampling Theory and Applications (SampTA) (2015), pp. 392-396 | DOI

[41] K. Pozharska; T. Ullrich A Note on Sampling Recovery of Multivariate Functions in the Uniform Norm, SIAM J. Numer. Anal., Volume 60 (2022) no. 3, pp. 1363-1384 | DOI | MR | Zbl

[42] H. Rauhut; R. Ward Sparse Legendre expansions via $ℓ_{1}$ -minimization, J. Approx. Theory, Volume 164 (2012) no. 5, pp. 517-533 | DOI | MR

[43] M. Rudelson; R. Vershynin Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab., Volume 18 (2013), 82, 9 pages | DOI | MR | Zbl

[44] J. Shen; T. Tang; L. Wang Spectral methods. Algorithms, analysis and applications, Springer Series in Computational Mathematics, 41, Springer, 2011, xvi+470 pages | DOI | MR

[45] I. Steinwart; A. Christmann Support vector machines, Information Science and Statistics, Springer, 2008, xvi+601 pages | MR

[46] G. Suryanarayana; D. Nuyens; R. Cools Reconstruction and collocation of a class of non-periodic functions by sampling along tent-transformed rank-1 lattices, J. Fourier Anal. Appl., Volume 22 (2016) no. 1, pp. 187-214 | DOI | MR | Zbl

[47] V. N. Temlyakov On approximate recovery of functions with bounded mixed derivative, J. Complexity, Volume 9 (1993) no. 1, pp. 41-59 (Festschrift for Joseph F. Traub, Part I) | DOI | MR | Zbl

[48] L. N. Trefethen Approximation theory and approximation practice, Society for Industrial and Applied Mathematics, 2013, viii+305 pages | DOI | MR

[49] H. Triebel Theory of function spaces, Modern Birkhäuser Classics, Birkhäuser, 2010, 285 pages | MR

[50] Hans Triebel Theory of function spaces. II, Monographs in Mathematics, 84, Birkhäuser, 1992, viii+370 pages | DOI | MR

[51] J. A. Tropp User-friendly tail bounds for sums of random matrices, Found. Comput. Math., Volume 12 (2012) no. 4, pp. 389-434 | DOI | MR | Zbl

[52] H. Wang New error bounds for Legendre approximations of differentiable functions (2021) (https://arxiv.org/abs/2111.03833) | DOI

[53] A. G. Werschulz; H. Woźniakowski Tractability of Multivariate Approximation over a Weighted Unanchored Sobolev Space, Constr. Approx., Volume 30 (2009) no. 3, pp. 395-421 | DOI | MR | Zbl

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