September  2020, 2(3): 279-307. doi: 10.3934/fods.2020013

Probabilistic learning on manifolds

1. 

Université Gustave Eiffel, Laboratoire Modélisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-la-Vallée, France

2. 

University of Southern California, Viterbi School of Engineering, 210 KAP Hall, Los Angeles, CA 90089, USA

* Corresponding author: Christian Soize

Published  August 2020

This paper presents novel mathematical results in support of the probabilistic learning on manifolds (PLoM) recently introduced by the authors. An initial dataset, constituted of a small number of points given in an Euclidean space, is given. The points are independent realizations of a vector-valued random variable for which its non-Gaussian probability measure is unknown but is, a priori, concentrated in an unknown subset of the Euclidean space. A learned dataset, constituted of additional realizations, is constructed. A transport of the probability measure estimated with the initial dataset is done through a linear transformation constructed using a reduced-order diffusion-maps basis. It is proven that this transported measure is a marginal distribution of the invariant measure of a reduced-order Itô stochastic differential equation. The concentration of the probability measure is preserved. This property is shown by analyzing a distance between the random matrix constructed with the PLoM and the matrix representing the initial dataset, as a function of the dimension of the basis. It is further proven that this distance has a minimum for a dimension of the reduced-order diffusion-maps basis that is strictly smaller than the number of points in the initial dataset.

Citation: Christian Soize, Roger Ghanem. Probabilistic learning on manifolds. Foundations of Data Science, 2020, 2 (3) : 279-307. doi: 10.3934/fods.2020013
References:
[1]

M. Arnst, C. Soize and K. Bulthies, Computation of Sobol indices in global sensitivity analysis from small data sets by probabilistic learning on manifolds, International Journal for Uncertainty Quantification, 1–34, online 18 August 2020. doi: 10.1615/Int.J.UncertaintyQuantification.2020032674.  Google Scholar

[2] A. Bowman and A. Azzalini, Applied Smoothing Techniques for Data Analysis: The Kernel Approach With S-Plus Illustrations, vol. 18, Oxford University Press, Oxford: Clarendon Press, New York, 1997.   Google Scholar
[3]

K. BurrageI. Lenane and G. Lythe, Numerical methods for second-order stochastic differential equations, SIAM J. Sci. Comput., 29 (2007), 245-264.  doi: 10.1137/050646032.  Google Scholar

[4]

R. R. Coifman and S. Lafon, Diffusion maps, Appl. Comput. Harmon. Anal., 21 (2006), 5-30.  doi: 10.1016/j.acha.2006.04.006.  Google Scholar

[5]

R. R. Coifman and S. Lafon, Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions, Appl. Comput. Harmon. Anal., 21 (2006), 31-52.  doi: 10.1016/j.acha.2005.07.005.  Google Scholar

[6]

R. R. CoifmanS. LafonA. B. LeeM. MaggioniB. NadlerF. Warner and S. W. Zucker, Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps, PNAS, 102 (2005), 7426-7431.  doi: 10.1073/pnas.0500334102.  Google Scholar

[7]

T. M. Cover and J. A. Thomas, Elements of Information Theory, Second Edition, John Wiley & Sons, Hoboken, NJ, 2006.  Google Scholar

[8]

J. L. Doob, Stochastic Processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953.  Google Scholar

[9]

T. DuongA. CowlingI. Koch and M. P. Wand, Feature significance for multivariate kernel density estimation, Comput. Statist. Data Anal., 52 (2008), 4225-4242.  doi: 10.1016/j.csda.2008.02.035.  Google Scholar

[10]

T. Duong and M. L. Hazelton, Cross-validation bandwidth matrices for multivariate kernel density estimation, Scand. J. Statist., 32 (2005), 485-506.  doi: 10.1111/j.1467-9469.2005.00445.x.  Google Scholar

[11]

C. FarhatR. TezaurT. ChapmanP. Avery and C. Soize, Feasible probabilistic learning method for model-form uncertainty quantification in vibration analysis, AIAA Journal, 57 (2019), 4978-4991.  doi: 10.2514/1.J057797.  Google Scholar

[12]

M. Filippone and G. Sanguinetti, Approximate inference of the bandwidth in multivariate kernel density estimation, Comput. Statist. Data Anal., 55 (2011), 3104-3122.  doi: 10.1016/j.csda.2011.05.023.  Google Scholar

[13]

R. Ghanem and C. Soize, Probabilistic nonconvex constrained optimization with fixed number of function evaluations, Internat. J. Numer. Methods Engrg., 113 (2018), 719-741.  doi: 10.1002/nme.5632.  Google Scholar

[14]

R. Ghanem, C. Soize, L. Mehrez and V. Aitharaju, Probabilistic learning and updating of a digital twin for composite material systems, International Journal for Numerical Methods in Engineering, 1–21. doi: 10.1002/nme.6430.  Google Scholar

[15]

R. GhanemC. Soize and C. Thimmisetty, Optimal well-placement using probabilistic learning, Data-Enabled Discovery and Applications, 2 (2018), 1-16.  doi: 10.1007/s41688-017-0014-x.  Google Scholar

[16]

R. G. Ghanem, C. Soize, C. Safta, X. Huan, G. Lacaze, J. C. Oefelein and H. N. Najm, Design optimization of a scramjet under uncertainty using probabilistic learning on manifolds, J. Comput. Phys., 399 (2019), 108930, 14 pp. doi: 10.1016/j.jcp.2019.108930.  Google Scholar

[17]

M. Girolami and B. Calderhead, Riemann manifold Langevin and Hamiltonian Monte Carlo methods, J. R. Stat. Soc. Ser. B Stat. Methodol., 73 (2011), 123-214.  doi: 10.1111/j.1467-9868.2010.00765.x.  Google Scholar

[18]

J. Guilleminot and J. E. Dolbow, Data-driven enhancement of fracture paths in random composites, Mechanics Research Communications, 103 (2020), 103443, 1–12. doi: 10.1016/j.mechrescom.2019.103443.  Google Scholar

[19]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition. Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2006.  Google Scholar

[20]

E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev., 106 (1957), 620-630.  doi: 10.1103/PhysRev.106.620.  Google Scholar

[21]

E. T. Jaynes, Information theory and statistical mechanics. ii, Phys. Rev., 108 (1957), 171-190.  doi: 10.1103/PhysRev.108.171.  Google Scholar

[22]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160. Springer-Verlag, New York, 2005.  Google Scholar

[23]

R. Z. Khasminskiǐ, Stochastic Stability of Differential Equations, vol. 66, Translated from the Russian by D. Louvish. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7. Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980.  Google Scholar

[24]

P. Krée and C. Soize, Mathematics of Random Phenomena, Random vibrations of mechanical structures. Translated from the French by Andrei Iacob. With a preface by Paul Germain. Mathematics and its Applications, 32. D. Reidel Publishing Co., Dordrecht, 1986. doi: 10.1007/978-94-009-4770-2.  Google Scholar

[25]

S. Lafon and A. B. Lee, Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning, and data set parameterization, IEEE Transactions on Pattern Analysis and Machine Intelligence, 28 (2006), 1393-1403.  doi: 10.1109/TPAMI.2006.184.  Google Scholar

[26]

R. M. Neal, MCMC using Hamiltonian dynamics, in Handbook of Markov Chain Monte Carlo, 113-162, Chapman & Hall/CRC Handb. Mod. Stat. Methods, CRC Press, Boca Raton, FL, 2011.  Google Scholar

[27]

E. Parzen, On estimation of a probability density function and mode, Ann. Math. Statist., 33 (1962), 1065-1076.  doi: 10.1214/aoms/1177704472.  Google Scholar

[28]

C. P. Robert and G. Casella, Monte Carlo Statistical Methods, Second edition. Springer Texts in Statistics. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4145-2.  Google Scholar

[29]

C. E. Shannon, A mathematical theory of communication, Bell System Tech. J., 27 (1948), 379–423 & 623–656. doi: 10.1002/j.1538-7305.1948.tb01338.x.  Google Scholar

[30]

C. Soize, The Fokker-Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions, Series on Advances in Mathematics for Applied Sciences, 17. World Scientific Publishing Co., Inc., River Edge, NJ, 1994. doi: 10.1142/9789814354110.  Google Scholar

[31]

C. Soize, Construction of probability distributions in high dimension using the maximum entropy principle. applications to stochastic processes, random fields and random matrices, Internat. J. Numer. Methods Engrg., 76 (2008), 1583-1611.  doi: 10.1002/nme.2385.  Google Scholar

[32]

C. Soize, Design optimization under uncertainties of a mesoscale implant in biological tissues using a probabilistic learning algorithm, Comput. Mech., 62 (2018), 477-497.  doi: 10.1007/s00466-017-1509-x.  Google Scholar

[33]

C. Soize and C. Farhat, Probabilistic learning for modeling and quantifying model-form uncertainties in nonlinear computational mechanics, Internat. J. Numer. Methods Engrg., 117 (2019), 819-843.  doi: 10.1002/nme.5980.  Google Scholar

[34]

C. Soize and R. Ghanem, Data-driven probability concentration and sampling on manifold, J. Comput. Phys., 321 (2016), 242-258.  doi: 10.1016/j.jcp.2016.05.044.  Google Scholar

[35]

C. Soize and R. Ghanem, Physics-constrained non-Gaussian probabilistic learning on manifolds, Internat. J. Numer. Methods Engrg., 121 (2020), 110-145.  doi: 10.1002/nme.6202.  Google Scholar

[36]

C. Soize, R. G. Ghanem and C. Desceliers, Sampling of Bayesian posteriors with a non-Gaussian probabilistic learning on manifolds from a small dataset, Statistics and Computing, 1–25 and Supplementary Material, 1–13, on line 08 June, 2020. doi: 10.1007/s11222-020-09954-6.  Google Scholar

[37]

C. SoizeR. GhanemC. SaftaX. HuanZ. P. VaneJ. C. OefeleinG. LacazeH. N. NajmQ. Tang and X. Chen, Entropy-based closure for probabilistic learning on manifolds, J. Comput. Phys., 388 (2019), 518-533.  doi: 10.1016/j.jcp.2018.12.029.  Google Scholar

[38]

J. C. Spall, Introduction to Stochastic Searh and Optimization, Wiley-Interscience, 2003. doi: 10.1002/0471722138.  Google Scholar

[39]

N. G. Trillos, F. Hoffmann and B. Hosseini, Geometric structure of graph laplacian embeddings, arXiv: 1901.10651, (2019). Google Scholar

[40]

N. ZougabS. Adjabi and C. C. Kokonendji, Bayesian estimation of adaptive bandwidth matrices in multivariate kernel density estimation, Comput. Statist. Data Anal., 75 (2014), 28-38.  doi: 10.1016/j.csda.2014.02.002.  Google Scholar

show all references

References:
[1]

M. Arnst, C. Soize and K. Bulthies, Computation of Sobol indices in global sensitivity analysis from small data sets by probabilistic learning on manifolds, International Journal for Uncertainty Quantification, 1–34, online 18 August 2020. doi: 10.1615/Int.J.UncertaintyQuantification.2020032674.  Google Scholar

[2] A. Bowman and A. Azzalini, Applied Smoothing Techniques for Data Analysis: The Kernel Approach With S-Plus Illustrations, vol. 18, Oxford University Press, Oxford: Clarendon Press, New York, 1997.   Google Scholar
[3]

K. BurrageI. Lenane and G. Lythe, Numerical methods for second-order stochastic differential equations, SIAM J. Sci. Comput., 29 (2007), 245-264.  doi: 10.1137/050646032.  Google Scholar

[4]

R. R. Coifman and S. Lafon, Diffusion maps, Appl. Comput. Harmon. Anal., 21 (2006), 5-30.  doi: 10.1016/j.acha.2006.04.006.  Google Scholar

[5]

R. R. Coifman and S. Lafon, Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions, Appl. Comput. Harmon. Anal., 21 (2006), 31-52.  doi: 10.1016/j.acha.2005.07.005.  Google Scholar

[6]

R. R. CoifmanS. LafonA. B. LeeM. MaggioniB. NadlerF. Warner and S. W. Zucker, Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps, PNAS, 102 (2005), 7426-7431.  doi: 10.1073/pnas.0500334102.  Google Scholar

[7]

T. M. Cover and J. A. Thomas, Elements of Information Theory, Second Edition, John Wiley & Sons, Hoboken, NJ, 2006.  Google Scholar

[8]

J. L. Doob, Stochastic Processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953.  Google Scholar

[9]

T. DuongA. CowlingI. Koch and M. P. Wand, Feature significance for multivariate kernel density estimation, Comput. Statist. Data Anal., 52 (2008), 4225-4242.  doi: 10.1016/j.csda.2008.02.035.  Google Scholar

[10]

T. Duong and M. L. Hazelton, Cross-validation bandwidth matrices for multivariate kernel density estimation, Scand. J. Statist., 32 (2005), 485-506.  doi: 10.1111/j.1467-9469.2005.00445.x.  Google Scholar

[11]

C. FarhatR. TezaurT. ChapmanP. Avery and C. Soize, Feasible probabilistic learning method for model-form uncertainty quantification in vibration analysis, AIAA Journal, 57 (2019), 4978-4991.  doi: 10.2514/1.J057797.  Google Scholar

[12]

M. Filippone and G. Sanguinetti, Approximate inference of the bandwidth in multivariate kernel density estimation, Comput. Statist. Data Anal., 55 (2011), 3104-3122.  doi: 10.1016/j.csda.2011.05.023.  Google Scholar

[13]

R. Ghanem and C. Soize, Probabilistic nonconvex constrained optimization with fixed number of function evaluations, Internat. J. Numer. Methods Engrg., 113 (2018), 719-741.  doi: 10.1002/nme.5632.  Google Scholar

[14]

R. Ghanem, C. Soize, L. Mehrez and V. Aitharaju, Probabilistic learning and updating of a digital twin for composite material systems, International Journal for Numerical Methods in Engineering, 1–21. doi: 10.1002/nme.6430.  Google Scholar

[15]

R. GhanemC. Soize and C. Thimmisetty, Optimal well-placement using probabilistic learning, Data-Enabled Discovery and Applications, 2 (2018), 1-16.  doi: 10.1007/s41688-017-0014-x.  Google Scholar

[16]

R. G. Ghanem, C. Soize, C. Safta, X. Huan, G. Lacaze, J. C. Oefelein and H. N. Najm, Design optimization of a scramjet under uncertainty using probabilistic learning on manifolds, J. Comput. Phys., 399 (2019), 108930, 14 pp. doi: 10.1016/j.jcp.2019.108930.  Google Scholar

[17]

M. Girolami and B. Calderhead, Riemann manifold Langevin and Hamiltonian Monte Carlo methods, J. R. Stat. Soc. Ser. B Stat. Methodol., 73 (2011), 123-214.  doi: 10.1111/j.1467-9868.2010.00765.x.  Google Scholar

[18]

J. Guilleminot and J. E. Dolbow, Data-driven enhancement of fracture paths in random composites, Mechanics Research Communications, 103 (2020), 103443, 1–12. doi: 10.1016/j.mechrescom.2019.103443.  Google Scholar

[19]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition. Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2006.  Google Scholar

[20]

E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev., 106 (1957), 620-630.  doi: 10.1103/PhysRev.106.620.  Google Scholar

[21]

E. T. Jaynes, Information theory and statistical mechanics. ii, Phys. Rev., 108 (1957), 171-190.  doi: 10.1103/PhysRev.108.171.  Google Scholar

[22]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160. Springer-Verlag, New York, 2005.  Google Scholar

[23]

R. Z. Khasminskiǐ, Stochastic Stability of Differential Equations, vol. 66, Translated from the Russian by D. Louvish. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7. Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980.  Google Scholar

[24]

P. Krée and C. Soize, Mathematics of Random Phenomena, Random vibrations of mechanical structures. Translated from the French by Andrei Iacob. With a preface by Paul Germain. Mathematics and its Applications, 32. D. Reidel Publishing Co., Dordrecht, 1986. doi: 10.1007/978-94-009-4770-2.  Google Scholar

[25]

S. Lafon and A. B. Lee, Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning, and data set parameterization, IEEE Transactions on Pattern Analysis and Machine Intelligence, 28 (2006), 1393-1403.  doi: 10.1109/TPAMI.2006.184.  Google Scholar

[26]

R. M. Neal, MCMC using Hamiltonian dynamics, in Handbook of Markov Chain Monte Carlo, 113-162, Chapman & Hall/CRC Handb. Mod. Stat. Methods, CRC Press, Boca Raton, FL, 2011.  Google Scholar

[27]

E. Parzen, On estimation of a probability density function and mode, Ann. Math. Statist., 33 (1962), 1065-1076.  doi: 10.1214/aoms/1177704472.  Google Scholar

[28]

C. P. Robert and G. Casella, Monte Carlo Statistical Methods, Second edition. Springer Texts in Statistics. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4145-2.  Google Scholar

[29]

C. E. Shannon, A mathematical theory of communication, Bell System Tech. J., 27 (1948), 379–423 & 623–656. doi: 10.1002/j.1538-7305.1948.tb01338.x.  Google Scholar

[30]

C. Soize, The Fokker-Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions, Series on Advances in Mathematics for Applied Sciences, 17. World Scientific Publishing Co., Inc., River Edge, NJ, 1994. doi: 10.1142/9789814354110.  Google Scholar

[31]

C. Soize, Construction of probability distributions in high dimension using the maximum entropy principle. applications to stochastic processes, random fields and random matrices, Internat. J. Numer. Methods Engrg., 76 (2008), 1583-1611.  doi: 10.1002/nme.2385.  Google Scholar

[32]

C. Soize, Design optimization under uncertainties of a mesoscale implant in biological tissues using a probabilistic learning algorithm, Comput. Mech., 62 (2018), 477-497.  doi: 10.1007/s00466-017-1509-x.  Google Scholar

[33]

C. Soize and C. Farhat, Probabilistic learning for modeling and quantifying model-form uncertainties in nonlinear computational mechanics, Internat. J. Numer. Methods Engrg., 117 (2019), 819-843.  doi: 10.1002/nme.5980.  Google Scholar

[34]

C. Soize and R. Ghanem, Data-driven probability concentration and sampling on manifold, J. Comput. Phys., 321 (2016), 242-258.  doi: 10.1016/j.jcp.2016.05.044.  Google Scholar

[35]

C. Soize and R. Ghanem, Physics-constrained non-Gaussian probabilistic learning on manifolds, Internat. J. Numer. Methods Engrg., 121 (2020), 110-145.  doi: 10.1002/nme.6202.  Google Scholar

[36]

C. Soize, R. G. Ghanem and C. Desceliers, Sampling of Bayesian posteriors with a non-Gaussian probabilistic learning on manifolds from a small dataset, Statistics and Computing, 1–25 and Supplementary Material, 1–13, on line 08 June, 2020. doi: 10.1007/s11222-020-09954-6.  Google Scholar

[37]

C. SoizeR. GhanemC. SaftaX. HuanZ. P. VaneJ. C. OefeleinG. LacazeH. N. NajmQ. Tang and X. Chen, Entropy-based closure for probabilistic learning on manifolds, J. Comput. Phys., 388 (2019), 518-533.  doi: 10.1016/j.jcp.2018.12.029.  Google Scholar

[38]

J. C. Spall, Introduction to Stochastic Searh and Optimization, Wiley-Interscience, 2003. doi: 10.1002/0471722138.  Google Scholar

[39]

N. G. Trillos, F. Hoffmann and B. Hosseini, Geometric structure of graph laplacian embeddings, arXiv: 1901.10651, (2019). Google Scholar

[40]

N. ZougabS. Adjabi and C. C. Kokonendji, Bayesian estimation of adaptive bandwidth matrices in multivariate kernel density estimation, Comput. Statist. Data Anal., 75 (2014), 28-38.  doi: 10.1016/j.csda.2014.02.002.  Google Scholar

Figure 1.  Left figure: for $ \varepsilon_ {\hbox{DM}} = \varepsilon_ {\hbox{opt}} $, distribution of the eigenvalues $ \lambda_\alpha(\varepsilon_ {\hbox{opt}}) $ in log scale as a function of rank $ \alpha $. Right figure: graph of function $ m\mapsto \varepsilon_d(m) $
Figure 2.  Left figure: distribution of the eigenvalues $ \lambda_\alpha(\varepsilon_ {\hbox{opt}}) $ in log scale as a function of rank $ \alpha\leq 50 $ for $ \varepsilon_ {\hbox{DM}} = \varepsilon_ {\hbox{opt}} = 60 $. Right figure: graph of function $ m\mapsto \varepsilon_d(m) $ for $ m\leq 50 $
Figure 3.  Left figure: graph of function $ m \mapsto f_d(m) $. Right figure: graph of function $ m\mapsto \underline g(m) $
Figure 4.  Left figure: graph of function $ m \mapsto d_N^{2, {\hbox{sim}}}(m) $. Right figure: graph of function $ m \mapsto d_N^{2, {\hbox{sim}}}(m) $ (blue dashed line), and for $ m\geq m_ {\hbox{opt}} $, graphs of $ m \mapsto d_N^{2,c}(m) $ (dark thick straight line) and $ m \mapsto d_N^{2, {\hbox{app}}}(m) $ (red thick curve line)
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