Convergence of a normed eigenvector stochastic approximation process and application to online principal component analysis of a data stream

Jean-Marie Monnez (1, 2, 3, 4) , Abderrahman Skiredj (5, 2)

1 CIC-P - Centre d'investigation clinique plurithématique Pierre Drouin [Nancy]
2 IECL - Institut Élie Cartan de Lorraine
3 IUT Nancy Charlemagne
4 BIGS - Biology, genetics and statistics
5 ENSMN - Ecole Nationale Supérieure des Mines de Nancy

Many articles were devoted to the problem of estimating recursively the eigenvectors and eigenvalues in decreasing order of the expectation of a random matrix using an i.i.d. sample of it. The present study makes the following contributions. The convergence of a normed process is proved under more general assumptions: the random matrices are not supposed i.i.d. and a new data mini-batch or all data until the current step are taken into account at each step without storing them; three types of processes are studied; this is applied to online principal component analysis of a data stream, assuming that data are realizations of a random vector Z whose expectation is unknown and must be estimated online, as well as possibly the metrics used when it depends on unknown characteristics of Z.

Mots clés: Stochastic approximation algorithms; Big Data; Principal Component Analysis; Online Estimation; Data Stream; Stochastic Ap- proximation

Domaines: Statistiques [math.ST]; Probabilités [math.PR]

Jean-Marie Monnez, Abderrahman Skiredj. Convergence of a normed eigenvector stochastic approximation process and application to online principal component analysis of a data stream. [Research Report] IECL. 2019. ⟨hal-01844419v2⟩