Teixeira Parente, Mario; Wallin, Jonas; Wohlmuth, Barbara Generalized bounds for active subspaces. (English) Zbl 1434.60059 Electron. J. Stat. 14, No. 1, 917-943 (2020). Summary: In this article, we consider scenarios in which traditional estimates for the active subspace method based on probabilistic Poincaré inequalities are not valid due to unbounded Poincaré constants. Consequently, we propose a framework that allows to derive generalized estimates in the sense that it enables to control the trade-off between the size of the Poincaré constant and a weaker order of the final error bound. In particular, we investigate independently exponentially distributed random variables in dimension two or larger and give explicit expressions for corresponding Poincaré constants showing their dependence on the dimension of the problem. Finally, we suggest possibilities for future work that aim for extending the class of distributions applicable to the active subspace method as we regard this as an opportunity to enlarge its usability. Cited in 8 Documents MSC: 60D05 Geometric probability and stochastic geometry 60-08 Computational methods for problems pertaining to probability theory Keywords:dimension reduction; active subspaces; Poincaré inequalities Software:Mathematica × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] Alonso-Gutiérrez, D. and Bastero, J. (2015)., Approaching the Kannan-Lovász-Simonovits and Variance Conjectures 2131. Springer. · Zbl 1318.52001 [2] Asar, Ö., Bolin, D., Diggle, P. J. and Wallin, J. (2018). Linear Mixed-Effects Models for Non-Gaussian Repeated Measurement Data., accepted for publication in Journal of the Royal Statistical Society: Series C. [3] Bakry, D., Gentil, I. and Ledoux, M. (2013)., Analysis and Geometry of Markov Diffusion Operators 348. Springer Science & Business Media. · Zbl 1376.60002 [4] Barndorff-Nielsen, O. (1977). Exponentially Decreasing Distributions for the Logarithm of Particle Size., Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 353 401-419. [5] Barndorff-Nielsen, O. E. (1997). Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling., Scandinavian Journal of statistics 24 1-13. · Zbl 0934.62109 · doi:10.1111/1467-9469.00045 [6] Barndorff-Nielsen, O. E. (1997). Processes of Normal Inverse Gaussian Type., Finance and stochastics 2 41-68. · Zbl 0894.90011 · doi:10.1007/s007800050032 [7] Bebendorf, M. (2003). A Note on the Poincaré Inequality for Convex Domains., Zeitschrift für Analysis und ihre Anwendungen 22 751-756. · Zbl 1057.26011 · doi:10.4171/ZAA/1170 [8] Bobkov, S. G. (1999). Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures., The Annals of Probability 27 1903-1921. · Zbl 0964.60013 [9] Bolin, D. (2014). Spatial Matérn Fields Driven by Non-Gaussian Noise., Scandinavian Journal of Statistics 41 557-579. · Zbl 1309.62158 · doi:10.1111/sjos.12046 [10] Bolin, D. and Wallin, J. (2019). Multivariate Type G Matérn Stochastic Partial Differential Equation Random Fields., Journal of the Royal Statistical Society: Series B (Statistical Methodology). · Zbl 1440.62180 [11] Bridges, R., Gruber, A., Felder, C., Verma, M. and Hoff, C. (2019). Active Manifolds: A Non-Linear Analogue to Active Subspaces. In, International Conference on Machine Learning 764-772. [12] Buchholz, R. H. (1992). Perfect Pyramids., Bulletin of the Australian Mathematical Society 45 353-368. · Zbl 0747.52008 · doi:10.1017/S0004972700030252 [13] Budninskiy, M., Yin, G., Feng, L., Tong, Y. and Desbrun, M. (2019). Parallel Transport Unfolding: A Connection-Based Manifold Learning Approach., SIAM Journal on Applied Algebra and Geometry 3 266-291. · Zbl 1425.53019 · doi:10.1137/18M1196133 [14] Chen, L. H. (1982). An Inequality for the Multivariate Normal Distribution., Journal of Multivariate Analysis 12 306-315. · Zbl 0483.60011 · doi:10.1016/0047-259X(82)90022-7 [15] Coleman, K. D., Lewis, A., Smith, R. C., Williams, B., Morris, M. and Khuwaileh, B. (2019). Gradient-Free Construction of Active Subspaces for Dimension Reduction in Complex Models with Applications to Neutronics., SIAM/ASA Journal on Uncertainty Quantification 7 117-142. · Zbl 1453.62561 · doi:10.1137/16M1075119 [16] Constantine, P. and Gleich, D. (2014). Computing Active Subspaces with Monte Carlo. arXiv preprint, arXiv:1408.0545. [17] Constantine, P. G. (2015)., Active Subspaces. SIAM Spotlights 2. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA Emerging Ideas for Dimension Reduction in Parameter Studies. · Zbl 1431.65001 [18] Constantine, P. G., Dow, E. and Wang, Q. (2014). Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces., SIAM J. Sci. Comput. 36 A1500-A1524. · Zbl 1311.65008 [19] Constantine, P. G., Eftekhari, A., Hokanson, J. and Ward, R. A. (2017). A Near-Stationary Subspace for Ridge Approximation., Computer Methods in Applied Mechanics and Engineering 326 402-421. · Zbl 1439.65023 · doi:10.1016/j.cma.2017.07.038 [20] Constantine, P. G., Kent, C. and Bui-Thanh, T. (2016). Accelerating Markov chain Monte Carlo with Active Subspaces., SIAM J. Sci. Comput. 38 A2779-A2805. · Zbl 1348.65010 · doi:10.1137/15M1042127 [21] Cui, T., Martin, J., Marzouk, Y. M., Solonen, A. and Spantini, A. (2014). Likelihood-Informed Dimension Reduction for Nonlinear Inverse Problems., Inverse Problems 30 114015, 28. · Zbl 1310.62030 · doi:10.1088/0266-5611/30/11/114015 [22] Diaz, P., Constantine, P., Kalmbach, K., Jones, E. and Pankavich, S. (2018). A Modified SEIR Model for the Spread of Ebola in Western Africa and Metrics for Resource Allocation., Applied Mathematics and Computation 324 141-155. · Zbl 1426.92072 · doi:10.1016/j.amc.2017.11.039 [23] Eberlein, E. (2001). Application of Generalized Hyperbolic Lévy Motions to Finance. In, Lévy processes 319-336. Springer. · Zbl 0982.60045 [24] Glaws, A. and Constantine, P. G. (2018). A Lanczos-Stieltjes Method for One-Dimensional Ridge Function Approximation and Integration. arXiv preprint, arXiv:1808.02095. [25] Hokanson, J. M. and Constantine, P. G. (2018). Data-Driven Polynomial Ridge Approximation Using Variable Projection., SIAM Journal on Scientific Computing 40 A1566-A1589. · Zbl 1392.49034 · doi:10.1137/17M1117690 [26] Jorgensen, B. (2012)., Statistical Properties of the Generalized Inverse Gaussian Distribution 9. Springer Science & Business Media. [27] Kinsey, L. C. and Moore, T. E. (2006)., Symmetry, Shape and Space: An Introduction to Mathematics Through Geometry. Springer Science & Business Media. · Zbl 0974.00002 [28] Klenke, A. (2013)., Probability Theory: A Comprehensive Course. Springer Science & Business Media. · Zbl 1141.60001 [29] Kotz, S., Kozubowski, T. and Podgorski, K. (2012)., The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance. Springer Science & Business Media. · Zbl 0977.62003 [30] Lee, Y. T. and Vempala, S. S. (2018). The Kannan-Lovász-Simonovits Conjecture. arXiv preprint, arXiv:1807.03465. · Zbl 1474.52005 [31] Leon, L. S., Smith, R. C., Oates, W. S. and Miles, P. (2017). Identifiability and Active Subspace Analysis for a Polydomain Ferroelectric Phase Field Model. In, ASME 2017 Conference on Smart Materials, Adaptive Structures and Intelligent Systems V002T03A022-V002T03A022. American Society of Mechanical Engineers. [32] Lin, T. I. and Lee, J. C. (2007). Bayesian Analysis of Hierarchical Linear Mixed Modeling Using the Multivariate, t Distribution. Journal of Statistical Planning and Inference 137 484-495. · Zbl 1102.62026 · doi:10.1016/j.jspi.2005.12.010 [33] Morawiec, A. (2003)., Orientations and Rotations. Springer. · Zbl 1084.74002 [34] Pinkus, A. (2015)., Ridge Functions 205. Cambridge University Press. · Zbl 1331.41001 [35] Stuart, A. M. (2010). Inverse Problems: A Bayesian Perspective., Acta Numerica 19 451-559. · Zbl 1242.65142 · doi:10.1017/S0962492910000061 [36] Teixeira Parente, M., Mattis, S., Gupta, S., Deusner, C. and Wohlmuth, B. (2019). Efficient Parameter Estimation for a Methane Hydrate Model with Active Subspaces., Computational Geosciences 23 355-372. · Zbl 1414.86004 · doi:10.1007/s10596-018-9769-x [37] Teixeira Parente, M., Bittner, D., Mattis, S. A., Chiogna, G. and Wohlmuth, B. (2019). Bayesian Calibration and Sensitivity Analysis for a Karst Aquifer Model Using Active Subspaces., Water Resources Research 55 7086-7107. [38] Tenenbaum, J. B., De Silva, V. and Langford, J. C. (2000). A Global Geometric Framework for Nonlinear Dimensionality Reduction., Science 290 2319-2323. [39] Tezzele, M., Ballarin, F. and Rozza, G. (2018). Combined Parameter and Model Reduction of Cardiovascular Problems by Means of Active Subspaces and POD-Galerkin Methods. In, Mathematical and Numerical Modeling of the Cardiovascular System and Applications 185-207. Springer. [40] Tripathy, R., Bilionis, I. and Gonzalez, M. (2016). Gaussian Processes with Built-In Dimensionality Reduction: Applications to High-Dimensional Uncertainty Propagation., Journal of Computational Physics 321 191-223. · Zbl 1349.65049 · doi:10.1016/j.jcp.2016.05.039 [41] van Handel, R. (2016). Probability in High Dimension Technical Report, Princeton, University. [42] Wallin, J. and Bolin, D. (2015). Geostatistical Modelling Using Non-Gaussian Matérn Fields., Scandinavian Journal of Statistics 42 872-890. · Zbl 1360.62517 · doi:10.1111/sjos.12141 [43] Wolfram Research, Inc. Mathematica, Version 11.3. Champaign, IL, 2018. [44] Yu, Y. (2017). On Normal Variance-Mean Mixtures., Statistics & Probability Letters 121 45-50. · Zbl 1375.60052 · doi:10.1016/j.spl.2016.07.024 [45] Zahm, O., Constantine, P., Prieur, C. and Marzouk, Y. (2018). Gradient-Based Dimension Reduction of Multivariate Vector-Valued Functions. arXiv preprint, arXiv:1801.07922. · Zbl 1433.41007 · doi:10.1137/18M1221837 [46] Zahm, O., Cui, T., Law, K., Spantini, A. and Marzouk, Y. (2018). Certified Dimension Reduction in Nonlinear Bayesian Inverse Problems. arXiv preprint, arXiv:1807.03712v2. · Zbl 07541892 [47] Zhang, P. · Zbl 1176.62018 · doi:10.1002/cjs.10015 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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