Widening the scope of an eigenvector stochastic approximation process and application to streaming PCA and related methods

We prove the almost sure convergence of Oja-type processes to eigenvectors of the expectation B of a random matrix while relaxing the i.i.d.assumption on the observed random matrices (Bn) and assuming either (Bn) converges to B or (E[Bn|Tn]) converges to B where Tn is the sigma-field generated by the events before time n. As an application of this generalization, the online PCA of a random vector Z can be performed when there is a data stream of i.i.d.observations of Z, even when both the metric M used and the expectation of Z are unknown and estimated online. Moreover, in order to update the stochastic approximation process at each step, we are no longer bound to using only a mini-batch of observations of Z, but all previous observations up to the current step can be used without having to store them. This is useful not only when dealing with streaming data but also with Big Data as one can process the latter sequentially as a data stream. In addition the general framework of this process, unlike other algorithms in the literature, also covers the case of factorial methods related to PCA.