Abstract
In this short paper, a matrix perturbation bound on the eigenvalues 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 acoustic metamaterial filters) is briefly discussed, together with possible extensions.
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Notes
- 1.
It is common practice to apply PCA to centered (also called de-meaned) data matrices \(\mathbf {X}^{(c)}\), i.e., having the form \(\mathbf {X}^{(c)} \doteq \mathbf {X}-\mathbf {1}_m \bar{\mathbf {x}}'\), where \(\mathbf {1}_m \in \mathbf {R}^m\) denotes a column vector made of m ones, and \(\bar{\mathbf {x}}\in \mathbf {R}^n\) is a column vector whose elements are the averages of the corresponding columns of \(\mathbf {X}\). This does not change the quality of the results of the theoretical analysis, because, by linearity, the centered convex combination of two data matrices \(\mathbf {X}_1\) and \(\mathbf {X}_2\) is equal to the convex combination of the two respective centered data matrices \(\mathbf {X}_1^{(c)}\) and \(\mathbf {X}_2^{(c)}\).
- 2.
- 3.
Such optimization problems are typically characterized by a high computational effort needed for an exact evaluation of the gradient of their objective functions, which is motivated by the fact that each such evaluation requires solving the physical-mathematical model associated with the specific choice of the vector of parameters of the model, which is also the vector of optimization variables.
- 4.
The reader is referred to [1] for examples of both single-objective and multi-objective optimal design problems for acoustic metamaterial filters (possible objective functions being the band gap and the band amplitude).
References
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Acknowledgment
A. Bacigalupo and G. Gnecco are members of INdAM. The authors acknowledge financial support from INdAM-GNAMPA (project Trade-off between Number of Examples and Precision in Variations of the Fixed-Effects Panel Data Model), from INdAM-GNFM, 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).
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Gnecco, G., Bacigalupo, A. (2022). On Principal Component Analysis of the Convex Combination of Two Data Matrices and Its Application to Acoustic Metamaterial Filters. In: Nicosia, G., et al. Machine Learning, Optimization, and Data Science. LOD 2021. Lecture Notes in Computer Science(), vol 13163. Springer, Cham. https://doi.org/10.1007/978-3-030-95467-3_9
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