Online high rank matrix completion
This work addresses matrix completion for high-rank data, which is incremental as it builds on existing kernel-based methods but introduces online capabilities for streaming data.
The paper tackles the problem of high rank matrix completion by mapping data into a high-dimensional polynomial feature space where it occupies a low-dimensional subspace, enabling efficient online and batch methods with reduced computational complexity compared to state-of-the-art approaches.
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.