  Recent advances in matrix completion enable data imputation in full-rank
matrices by exploiting low dimensional (nonlinear) latent structure. In this
paper, we develop a new model for high rank matrix completion (HRMC), together
with batch and online methods to fit the model and out-of-sample extension to
complete new data. The method works by (implicitly) mapping the data into a
high dimensional polynomial feature space using the kernel trick; importantly,
the data occupies a low dimensional subspace in this feature space, even when
the original data matrix is of full-rank. We introduce an explicit
parametrization of this low dimensional subspace, and an online fitting
procedure, to reduce computational complexity compared to the state of the art.
The online method can also handle streaming or sequential data and adapt to
non-stationary latent structure. We provide guidance on the sampling rate
required these methods to succeed. Experimental results on synthetic data and
motion capture data validate the performance of the proposed methods.
