  Extracting the underlying low-dimensional space where high-dimensional
signals often reside has long been at the center of numerous algorithms in the
signal processing and machine learning literature during the past few decades.
At the same time, working with incomplete (partly observed) large scale
datasets has recently been commonplace for diverse reasons. This so called {\it
big data era} we are currently living calls for devising online subspace
learning algorithms that can suitably handle incomplete data. Their envisaged
objective is to {\it recursively} estimate the unknown subspace by processing
streaming data sequentially, thus reducing computational complexity, while
obviating the need for storing the whole dataset in memory. In this paper, an
online variational Bayes subspace learning algorithm from partial observations
is presented. To account for the unawareness of the true rank of the subspace,
commonly met in practice, low-rankness is explicitly imposed on the sought
subspace data matrix by exploiting sparse Bayesian learning principles.
Moreover, sparsity, {\it simultaneously} to low-rankness, is favored on the
subspace matrix by the sophisticated hierarchical Bayesian scheme that is
adopted. In doing so, the proposed algorithm becomes adept in dealing with
applications whereby the underlying subspace may be also sparse, as, e.g., in
sparse dictionary learning problems. As shown, the new subspace tracking scheme
outperforms its state-of-the-art counterparts in terms of estimation accuracy,
in a variety of experiments conducted on simulated and real data.
