Supporting vectors vs. principal components

Almudena P. Márquez; Francisco Javier García-Pacheco; Míriam Mengibar-Rodríguez; Alberto Sánchez-Alzola; Almudena P. Márquez; Francisco Javier García-Pacheco; Míriam Mengibar-Rodríguez; Alberto Sánchez-Alzola

doi:10.3934/math.2023100

AIMS Mathematics

2023, Volume 8, Issue 1: 1937-1958. doi: 10.3934/math.2023100

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Research article Special Issues

Supporting vectors vs. principal components

1.
Department of Mathematics, College of Engineering, University of Cadiz, 11519 Puerto Real, Spain
2.
Department of Computer Science and Artificial Intelligence, University of Granada, 18071 Granada, Spain
3.
Department of Statistics and Operation Research, University of Cadiz, 11519 Puerto Real, Spain

Received: 10 July 2022 Revised: 29 September 2022 Accepted: 08 October 2022 Published: 26 October 2022
MSC : 51F30, 54E35, 54E45

Let $ T:X\to Y $ be a bounded linear operator between Banach spaces $ X, Y $. A vector $ x_0\in {\mathsf{S}}_X $ in the unit sphere $ {\mathsf{S}}_X $ of $ X $ is called a supporting vector of $ T $ provided that $ \|T(x_0)\| = \sup\{\|T(x)\|:\|x\| = 1\} = \|T\| $. Since matrices induce linear operators between finite-dimensional Hilbert spaces, we can consider their supporting vectors. In this manuscript, we unveil the relationship between the principal components of a matrix and its supporting vectors. Applications of our results to real-life problems are provided.
- bounded linear operator,
- Hilbert space,
- mean operator,
- principal components,
- supporting vector
Citation: Almudena P. Márquez, Francisco Javier García-Pacheco, Míriam Mengibar-Rodríguez, Alberto Sánchez-Alzola. Supporting vectors vs. principal components[J]. AIMS Mathematics, 2023, 8(1): 1937-1958. doi: 10.3934/math.2023100

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Abstract

Let $ T:X\to Y $ be a bounded linear operator between Banach spaces $ X, Y $. A vector $ x_0\in {\mathsf{S}}_X $ in the unit sphere $ {\mathsf{S}}_X $ of $ X $ is called a supporting vector of $ T $ provided that $ \|T(x_0)\| = \sup\{\|T(x)\|:\|x\| = 1\} = \|T\| $. Since matrices induce linear operators between finite-dimensional Hilbert spaces, we can consider their supporting vectors. In this manuscript, we unveil the relationship between the principal components of a matrix and its supporting vectors. Applications of our results to real-life problems are provided.

References

[1]	E. Bishop, R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc., 67 (1961), 97–98. https://doi.org/10.1090/S0002-9904-1961-10514-4 doi: 10.1090/S0002-9904-1961-10514-4
[2]	E. Bishop, R. R. Phelps, The support functionals of a convex set, In: Proceedings of Symposia in Pure Mathematics, Vol. VII, Providence, R.I.: Amer. Math. Soc., 1963, 27–35.
[3]	T. Bouwmans, S. Javed, H. Zhang, Z. Lin, R. Otazo, On the applications of robust PCA in image and video processing, Proc. IEEE, 106 (2018), 1427–1457. https://doi.org/10.1109/JPROC.2018.2853589 doi: 10.1109/JPROC.2018.2853589
[4]	C. Cobos-Sánchez, F. J. Garcia-Pacheco, J. M. Guerrero Rodriguez, J. R. Hill, An inverse boundary element method computational framework for designing optimal TMS coils, Eng. Anal. Bound. Elem., 88 (2018), 156–169. https://doi.org/10.1016/j.enganabound.2017.11.002 doi: 10.1016/j.enganabound.2017.11.002
[5]	C. Cobos-Sánchez, F. J. García-Pacheco, S. Moreno-Pulido, S. Sáez-Martínez, Supporting vectors of continuous linear operators, Ann. Funct. Anal., 8 (2017), 520–530. https://doi.org/10.1215/20088752-2017-0016 doi: 10.1215/20088752-2017-0016
[6]	C. Cobos-Sánchez, J. A. Vilchez-Membrilla, A. Campos-Jiménez, F. J. García-Pacheco, Pareto optimality for multioptimization of continuous linear operators, Symmetry, 13 (2021), 661. https://doi.org/10.3390/sym13040661 doi: 10.3390/sym13040661
[7]	C. Cobos-Sánchez, M. R. Cabello, Á. Q. Olozábal, M. F. Pantoja, Design of TMS coils with reduced lorentz forces: application to concurrent TMS-fMRI, J. Neural Eng., 17 (2020), 016056. https://doi.org/10.1088/1741-2552/ab4ba2 doi: 10.1088/1741-2552/ab4ba2
[8]	C. Cobos-Sánchez, J. J. J. García, M. R. Cabello, M. F. Pantoja, Design of coils for lateralized TMS on mice, J. Neural Eng., 17 (2020), 036007. https://doi.org/10.1088/1741-2552/ab89fe doi: 10.1088/1741-2552/ab89fe
[9]	C. Cobos-Sánchez, F. J. Garcia-Pacheco, J. M. Guerrero-Rodriguez, L. Garcia-Barrachina, Solving an IBEM with supporting vector analysis to design quiet TMS coils, Eng. Anal. Bound. Elem., 117 (2020), 1–12. https://doi.org/10.1016/j.enganabound.2020.04.013 doi: 10.1016/j.enganabound.2020.04.013
[10]	C. Cobos-Sánchez, J. M. Guerrero-Rodriguez, Á. Q. Olozábal, D. Blanco-Navarro, Novel TMS coils designed using an inverse boundary element method, Phys. Med. Biol., 62 (2016), 73–90. https://doi.org/10.1088/1361-6560/62/1/73 doi: 10.1088/1361-6560/62/1/73
[11]	C. M. Epstein, E. Wassermann, U. Ziemann, Oxford Handbook of Transcranial Stimulation, New York: Oxford University Press, 2008. https://doi.org/10.1093/oxfordhb/9780198568926.001.0001
[12]	J. Fan, Q. Sun, W.-X. Zhou, Z. Zhu, Principal component analysis for big data, Wiley StatsRef: Statistics Reference Online, in press. https://doi.org/10.1002/9781118445112.stat08122
[13]	F. J. García-Pacheco, E. Naranjo-Guerra, Supporting vectors of continuous linear projections, International Journal of Functional Analysis, Operator Theory and Applications, 9 (2017), 85–95.
[14]	F. J. García-Pacheco, Lineability of the set of supporting vectors, RACSAM, 115 (2021), 41, https://doi.org/10.1007/s13398-020-00981-6 doi: 10.1007/s13398-020-00981-6
[15]	F. J. Garcia-Pacheco, C. Cobos-Sánchez, S. Moreno-Pulido, A. Sanchez-Alzola, Exact solutions to $\max_{\Vert x\Vert = 1}\sum^\infty_{i = 1}\Vert T_i(x)\Vert^2$ with applications to Physics, Bioengineering and Statistics, Commun. Nonlinear Sci. Numer. Simul., 82 (2020), 105054. https://doi.org/10.1016/j.cnsns.2019.105054 doi: 10.1016/j.cnsns.2019.105054
[16]	F.-K. Garsiya-Pacheko, The cardinality of the set $\Lambda$ determines the geometry of the spaces $B_{\ell_\infty(\Lambda)}$ and $B_{\ell_\infty(\Lambda)^}$, (Russian), Funktsional. Anal. i Prilozhen.*, 52 (2018), 62–71. https://doi.org/10.4213/faa3534 doi: 10.4213/faa3534
[17]	Instituto de Estadística y Cartografía de Andalucía. Available from: https://www.juntadeandalucia.es/institutodeestadisticaycartografia.
[18]	R. C. James, Characterizations of reflexivity, Stud. Math., 23 (1964), 205–216. https://doi.org/10.4064/sm-23-3-205-216 doi: 10.4064/sm-23-3-205-216
[19]	J. Lindenstrauss, On operators which attain their norm, Israel J. Math., 1 (1963), 139–148. https://doi.org/10.1007/BF02759700 doi: 10.1007/BF02759700
[20]	L. Marin, H. Power, R. W. Bowtell, C. Cobos-Sánchez, A. A. Becker, P. Glover, et al., Numerical solution of an inverse problem in magnetic resonance imaging using a regularized higher-order boundary element method, In: Boundary elements and other mesh reduction methods XXIX, Southampton: WIT Press, 2007,323–332. https://doi.org/10.2495/BE070311
[21]	L. Marin, H. Power, R. W. Bowtell, C. Cobos-Sánchez, A. A. Becker, P. Glover, et al., Boundary element method for an inverse problem in magnetic resonance imaging gradient coils, CMES Comput. Model. Eng. Sci., 23 (2008), 149–173. https://doi.org/10.3970/cmes.2008.023.149 doi: 10.3970/cmes.2008.023.149
[22]	S. Moreno-Pulido, F. J. Garcia-Pacheco, C. Cobos-Sánchez, A. Sanchez-Alzola, Exact solutions to the maxmin problem $ \max \\|ax\\|$ subject to $\\|bx\\| \leq 1$, Mathematics, 8 (2020), 85. http://doi.org/10.3390/math8010085 doi: 10.3390/math8010085
[23]	A. Sánchez-Alzola, F. J. García-Pacheco, E. Naranjo-Guerra, S. Moreno-Pulido, Supporting vectors for the $\ell_1$-norm and the $\ell_\infty$-norm and an application, Math. Sci., 15 (2021), 173–187. https://doi.org/10.1007/s40096-021-00400-w doi: 10.1007/s40096-021-00400-w
[24]	L. Surhone, M. Timpledon, S. Marseken, Principal component analysis: Karhunen-Loève Theorem, Harold Hotelling, Karl Pearson, Exploratory Data Analysis, Eigendecomposition of a Matrix, Covariance Matrix, Singular Value Decomposition, Factor Analysis, Betascript Publishing, 2010.

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