# American Institute of Mathematical Sciences

doi: 10.3934/mfc.2021014
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## 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 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, doi: 10.3934/mfc.2021014
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(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)
]">Figure 2.  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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