Abstract
Numerous approximation methods have been developed to approximate both the kernel matrix and its inverse. We investigate one such influential approximation that has recently gained popularity. However, our results indicate that this approximation fails to address the ill-conditioning of the kernel matrix, potentially leading to significantly large biases and highly unstable prediction results.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Schölkopf, B., Smola, A.J.: Learning with Kernels. MIT Press, Cambridge, MA (2002)
Stein, M.L.: Interpolation of Spatial Data: Some Theory for Kriging. Springer (1999). https://doi.org/10.1007/978-1-4612-1494-6
Belkin, M.: Approximation beats concentration? An approximation view on inference with smooth radial kernels. Proc. Mach. Learn. Res. 75, 1–14 (2018)
Williams, C., Seeger, M.: Using the Nyström method to speed up kernel machines. In: Leen, T., Dietterich, T., Tresp, V. (eds.) Advances in Neural Information Processing Systems, vol. 13. MIT Press, Cambridge (2000)
Drineas, P., Mahoney, M.W.: On the Nystrom method for approximating a gram matrix for improved kernel-based learning. J. Mach. Learn. Res. 6, 2153–2175 (2005)
Bach, F.: Sharp analysis of low-rank kernel matrix approximations. In: JMLR: Workshop and Conference Proceedings, vol. 30, pp. 1–25, May 2013
Zhang, K., Tsang, I.W., Kwok, J.T.: Improved Nystrom low-rank approximation and error analysis. In: Proceedings of the 25th International Conference on Machine Learning, pp. 1232–1239. ACM (2008)
Gittens, A., Mahoney, M.W.: Revisiting the Nyström method for improved large-scale machine learning. J. Mach. Learn. Res. 17(1), 3977–4041 (2016)
Furrer, R., Genton, M.G., Nychka, D.: Covariance tapering for interpolation of large spatial datasets. J. Comput. Graph. Stat. 15(3), 502–523 (2006)
Kaufman, C.G., Schervish, M.J., Nychka, D.W.: Covariance tapering for likelihood-based estimation in large spatial data sets. J. Am. Stat. Assoc. 103(484), 1545–1555 (2008). Taylor & Francis
Stein, M.L., Chen, J., Anitescu, M., et al.: Stochastic approximation of score functions for Gaussian processes. Ann. Appl. Stat. 7(2), 1162–1191 (2013). Institute of Mathematical Statistics
Du, J., Zhang, H., Mandrekar, V.S.: Fixed-domain asymptotic properties of tapered maximum likelihood estimators. Ann. Stat. 37(6A), 3330–3361 (2009)
Stein, M.L., Chi, Z., Welty, L.J.: Approximating likelihoods for large spatial data sets. J. Roy. Stat. Soc. Ser. B (Stat. Methodol.) 66(2), 275–296 (2004). Wiley
Eidsvik, J., Shaby, B.A., Reich, B.J., Wheeler, M., Niemi, J.: Estimation and prediction in spatial models with block composite likelihoods. J. Comput. Graph. Stat. 23(2), 295–315 (2014). Taylor & Francis
Datta, A., Banerjee, S., Finley, A.O., Gelfand, A.E.: Hierarchical nearest-neighbor gaussian process models for large geostatistical datasets. J. Am. Stat. Assoc. 111(514), 800–812 (2016)
Guinness, J.: Permutation and grou** methods for sharpening gaussian process approximations. Technometrics 60(4), 415–429 (2018)
Datta, A.: Sparse nearest neighbor Cholesky matrices in spatial statistics (2021). ar**v:2102.13299 [stat]
Datta, A.: Nearest-neighbor sparse Cholesky matrices in spatial statistics. WIREs Comput. Stat. 14(5), e1574 (2022)
Zhang, H.: Spatial process approximations: assessing their necessity (2023). ar**v:2311.03201 [stat.ML]
Braun, M.L.: Accurate error bounds for the eigenvalues of the kernel matrix. J. Mach. Learn. Res. 7(82), 2303–2328 (2006)
Jia, L., Liao, S.: Accurate probabilistic error bound for eigenvalues of kernel matrix. In: Zhou, Z.-H., Washio, T. (eds.) ACML 2009. LNCS (LNAI), vol. 5828, pp. 162–175. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-05224-8_14
Vecchia, A.V.: Estimation and model identification for continuous spatial processes. J. Roy. Stat. Soc. B 50, 297–312 (1988)
Golub, G.H., Loan, C.F.V.: Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore (2012)
Trefethen, L.N., Bau III, D.: Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM) (1997)
O’Dowd, R.: Conditioning of coefficient matrices of ordinary kriging. Math. Geol. 23, 721–739 (1991)
McCourt, M., Fasshauer, G.E.: Stable likelihood computation for gaussian random fields. In: Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science: Novel Methods in Harmonic Analysis, vol. 2, pp. 917–943 (2017)
Basak, S., Petit, S., Bect, J., Vazquez, E.: Numerical issues in maximum likelihood parameter estimation for Gaussian process interpolation. In: Nicosia, G., et al. (eds.) LOD 2021. LNCS, vol. 13164, pp. 116–131. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-95470-3_9
Zhang, H.: Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics. J. Am. Stat. Assoc. 99(465), 250–261 (2004)
Wang, D., Loh, W.L.: On fixed-domain asymptotics and covariance tapering in Gaussian random field models. Electron. J. Statist 5, 238–269 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Chen, BY., Zhang, H. (2024). Large-Scale Data Challenges: Instability in Statistical Learning. In: Huang, DS., Premaratne, P., Yuan, C. (eds) Applied Intelligence. ICAI 2023. Communications in Computer and Information Science, vol 2015. Springer, Singapore. https://doi.org/10.1007/978-981-97-0827-7_17
Download citation
DOI: https://doi.org/10.1007/978-981-97-0827-7_17
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-97-0826-0
Online ISBN: 978-981-97-0827-7
eBook Packages: Computer ScienceComputer Science (R0)