# American Institute of Mathematical Sciences

November  2021, 4(4): 253-269. doi: 10.3934/mfc.2021014

## Convex combination of data matrices: PCA perturbation bounds for multi-objective optimal design of mechanical metafilters

 1 IMT School for Advanced Studies, AXES Research Unit, Piazza S. Francesco, 19, 55100 Lucca, Italy 2 University of Genoa, Department of Civil, Chemical and Environmental Engineering, Via Montallegro, 1, 16145 Genova, Italy

* Corresponding author: Giorgio Gnecco

Received  April 2021 Revised  July 2021 Published  November 2021 Early access  August 2021

Fund Project: A. Bacigalupo and G. Gnecco are members of INdAM. The authors acknowledge financial support from INdAM-GNAMPA, from INdAM-GNFM (project Trade-off between Number of Examples and Precision in Variations of the Fixed-Effects Panel Data Model), from the Università Italo Francese (projects GALILEO 2019 no. G19-48 and GALILEO 2021 no. G21 89), from the Compagnia di San Paolo (project MINIERA no. I34I20000380007), and from the University of Trento (project UNMASKED 2020)

In the present study, matrix perturbation bounds on the eigenvalues and on the invariant subspaces found by principal component analysis is investigated, for the case in which the data matrix on which principal component analysis is performed is a convex combination of two data matrices. The application of the theoretical analysis to multi-objective optimization problems – e.g., those arising in the design of mechanical metamaterial filters – is also discussed, together with possible extensions.

Citation: Giorgio Gnecco, Andrea Bacigalupo. Convex combination of data matrices: PCA perturbation bounds for multi-objective optimal design of mechanical metafilters. Mathematical Foundations of Computing, 2021, 4 (4) : 253-269. doi: 10.3934/mfc.2021014
##### References:
 [1] O. B. Augusto, F. Bennis and S. Caro, Multiobjective optimization involving quadratic functions, Journal of Optimization, 2014 (2014). [2] A. Bacigalupo, G. Gnecco, M. Lepidi and L. Gambarotta, Computational design of innovative mechanical metafilters via adaptive surrogate-based optimization, Comput. Methods Appl. Mech. Engrg., 375 (2021), 22pp. doi: 10.1016/j.cma.2020.113623. [3] A. Bacigalupo, G. Gnecco, M. Lepidi and L. Gambarotta, Multi-objective optimal design of mechanical metafilters, Submitted, (2021). [4] A. Bacigalupo, G. Gnecco, M. Lepidi and L. Gambarotta, Machine-learning techniques for the optimal design of acoustic metamaterials, J. Optim. Theory Appl., 187 (2020), 630-653.  doi: 10.1007/s10957-019-01614-8. [5] T. Chartier, When Life Is Linear: From Computer Graphics to Bracketology, The Mathematical Association of America, 2015. doi: 10.5948/9781614446163. [6] Y. Collette and P. Siarry, Multiobjective Optimization: Principles and Case Studies, Decision Engineering. Springer-Verlag, Berlin, 2003. [7] G. Gnecco and A. Bacigalupo, On principal component analysis of the convex combination of two data matrices and its application to acoustic metamaterial filters, In Proceedings of the Seventh International Conference on Machine Learning, Optimization, and Data Science (LOD), Lecture Notes in Computer Science, Forthcoming, (2021). [8] G. Gnecco, A. Bacigalupo, F. Fantoni and D. Selvi, Principal component analysis applied to gradient fields in band gap optimization problems for metamaterials, In IProceedings of the Sixth International Conference on Metamaterials and Nanophotonics (METANANO), Forthcoming, (2021). [9] G. Gnecco and M. Sanguineti, Accuracy of suboptimal solutions to kernel principal component analysis, Comput. Optim. Appl., 42 (2009), 265-287.  doi: 10.1007/s10589-007-9108-y. [10] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.  doi: 10.1017/CBO9780511840371. [11] I. T. Jolliffe, Principal Component Analysis, Springer, 2002. [12] I. Y. Kim and O. L. de Weck, Adaptive weighted-sum method for bi-objective optimization: Pareto front generation, Structural and Multidisciplinary Optimization, 29 (2005), 149-158. [13] R. Mathar, G. Alirezaei, E. Balda and A. Behboodi, Fundamentals of Data Analytics: With a View to Machine Learning, Springer, 2020. doi: 10.1007/978-3-030-56831-3. [14] P. A. Ruud, An Introduction to Classical Econometric Theory, Oxford University Press, 2000. [15] J. Shawe-Taylor and N. Cristianini, Kernel Methods for Pattern Analysis, Cambridge University Press, 2004. [16] G. W. Stewart and J.-G. Sun, Matrix Perturbation Theory, Academic Press, 1990. [17] G. Tzimiropoulos, S. Zafeiriou and M. Pantic, Principal component analysis of image gradient orientations for face recognition, In Proceedings of the Ninth IEEE International Conference on Automatic Face & Gesture Recognition (FG), (2011), 553–558. [18] G. Tzimiropoulos, S. Zafeiriou and M. Pantic, Subspace learning from image gradient orientations, IEEE Transactions on Pattern Analysis and Machine Intelligence, 34 (2012), 2454-2466. [19] F. Vadalà, A. Bacigalupo, M. Lepidi and L. Gambarotta, Free and forced wave propagation in beam lattice metamaterials with viscoelastic resonators, International Journal of Mechanical Sciences, 193 (2021). [20] P.-Å. Wedin, Perturbation bounds in connection with singular value decomposition, Nordisk Tidskr. Informationsbehandling (BIT), 12 (1972), 99-111.  doi: 10.1007/bf01932678. [21] Y. Yu, T. Wang and R.-J. Samworth, A useful variant of the Davis-Kahan theorem for statisticians, Biometrika, 102 (2015), 315-323.  doi: 10.1093/biomet/asv008. [22] P. Zhu and A. V. Knyazev, Angles between subspaces and their tangents, J. Numer. Math., 21 (2013), 325-340.  doi: 10.1515/jnum-2013-0013.

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##### References:
 [1] O. B. Augusto, F. Bennis and S. Caro, Multiobjective optimization involving quadratic functions, Journal of Optimization, 2014 (2014). [2] A. Bacigalupo, G. Gnecco, M. Lepidi and L. Gambarotta, Computational design of innovative mechanical metafilters via adaptive surrogate-based optimization, Comput. Methods Appl. Mech. Engrg., 375 (2021), 22pp. doi: 10.1016/j.cma.2020.113623. [3] A. Bacigalupo, G. Gnecco, M. Lepidi and L. Gambarotta, Multi-objective optimal design of mechanical metafilters, Submitted, (2021). [4] A. Bacigalupo, G. Gnecco, M. Lepidi and L. Gambarotta, Machine-learning techniques for the optimal design of acoustic metamaterials, J. Optim. Theory Appl., 187 (2020), 630-653.  doi: 10.1007/s10957-019-01614-8. [5] T. Chartier, When Life Is Linear: From Computer Graphics to Bracketology, The Mathematical Association of America, 2015. doi: 10.5948/9781614446163. [6] Y. Collette and P. Siarry, Multiobjective Optimization: Principles and Case Studies, Decision Engineering. Springer-Verlag, Berlin, 2003. [7] G. Gnecco and A. Bacigalupo, On principal component analysis of the convex combination of two data matrices and its application to acoustic metamaterial filters, In Proceedings of the Seventh International Conference on Machine Learning, Optimization, and Data Science (LOD), Lecture Notes in Computer Science, Forthcoming, (2021). [8] G. Gnecco, A. Bacigalupo, F. Fantoni and D. Selvi, Principal component analysis applied to gradient fields in band gap optimization problems for metamaterials, In IProceedings of the Sixth International Conference on Metamaterials and Nanophotonics (METANANO), Forthcoming, (2021). [9] G. Gnecco and M. Sanguineti, Accuracy of suboptimal solutions to kernel principal component analysis, Comput. Optim. Appl., 42 (2009), 265-287.  doi: 10.1007/s10589-007-9108-y. [10] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.  doi: 10.1017/CBO9780511840371. [11] I. T. Jolliffe, Principal Component Analysis, Springer, 2002. [12] I. Y. Kim and O. L. de Weck, Adaptive weighted-sum method for bi-objective optimization: Pareto front generation, Structural and Multidisciplinary Optimization, 29 (2005), 149-158. [13] R. Mathar, G. Alirezaei, E. Balda and A. Behboodi, Fundamentals of Data Analytics: With a View to Machine Learning, Springer, 2020. doi: 10.1007/978-3-030-56831-3. [14] P. A. Ruud, An Introduction to Classical Econometric Theory, Oxford University Press, 2000. [15] J. Shawe-Taylor and N. Cristianini, Kernel Methods for Pattern Analysis, Cambridge University Press, 2004. [16] G. W. Stewart and J.-G. Sun, Matrix Perturbation Theory, Academic Press, 1990. [17] G. Tzimiropoulos, S. Zafeiriou and M. Pantic, Principal component analysis of image gradient orientations for face recognition, In Proceedings of the Ninth IEEE International Conference on Automatic Face & Gesture Recognition (FG), (2011), 553–558. [18] G. Tzimiropoulos, S. Zafeiriou and M. Pantic, Subspace learning from image gradient orientations, IEEE Transactions on Pattern Analysis and Machine Intelligence, 34 (2012), 2454-2466. [19] F. Vadalà, A. Bacigalupo, M. Lepidi and L. Gambarotta, Free and forced wave propagation in beam lattice metamaterials with viscoelastic resonators, International Journal of Mechanical Sciences, 193 (2021). [20] P.-Å. Wedin, Perturbation bounds in connection with singular value decomposition, Nordisk Tidskr. Informationsbehandling (BIT), 12 (1972), 99-111.  doi: 10.1007/bf01932678. [21] Y. Yu, T. Wang and R.-J. Samworth, A useful variant of the Davis-Kahan theorem for statisticians, Biometrika, 102 (2015), 315-323.  doi: 10.1093/biomet/asv008. [22] P. Zhu and A. V. Knyazev, Angles between subspaces and their tangents, J. Numer. Math., 21 (2013), 325-340.  doi: 10.1515/jnum-2013-0013.
(a) Positive eigenvalues $\lambda_i({\bf{G}}(\alpha))$ (green curves, $i = 1,\ldots,5$), their best lower bounds derived from the first inequalities in Eqs. (1a) and (1b) in Proposition 1 (blue curves) with $K = 50$, and their best upper bounds derived from the same inequalities, still with $K = 50$ (red curves); (b) for $K = 1$, $i = 1$, and each $\alpha \in [0,1]$: $\sin(\theta_{1,{\rm min}}(\alpha))$ (green curve), and smallest upper bound on it, based on the second to last inequalities in Eqs. (11a) and (11b) in Proposition 2 (blue curve)
Beam lattice metamaterials with viscoelastic resonators and their reference periodic cell [19]
Floquet-Bloch spectrum maximizing a low-frequency band gap of a mechanical metamaterial filter: (a) $3$-dimensional representation; (b) projection of the spectrum onto a vertical plane
Floquet-Bloch spectrum maximizing a high-frequency pass band of a mechanical metamaterial filter: (a) $3$-dimensional representation; (b) projection of the spectrum onto a vertical plane
Floquet-Bloch spectrum maximizing a trade-off between a low-frequency bang gap and a high-frequency pass band of a mechanical metamaterial filter: (a) $3$-dimensional representation; (b) projection of the spectrum onto a vertical plane
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