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We consider a challenging multi-label classification problem where bothfeature matrix $\X$ and label matrix $\Y$ have missing entries. An existingmethod concatenated $\X$ and $\Y$ as $[\X; \Y]$ and applied a matrix completion(MC) method to fill the missing entries, under the assumption that $[\X; \Y]$is of low-rank. However, since entries of $\Y$ take binary values in themulti-label setting, it is unlikely that $\Y$ is of low-rank. Moreover, suchassumption implies a linear relationship between $\X$ and $\Y$ which may nothold in practice. In this paper, we consider a latent matrix $\Z$ that producesthe probability $\sigma(Z_{ij})$ of generating label $Y_{ij}$, where$\sigma(\cdot)$ is nonlinear. Considering label correlation, we assume $[\X;\Z]$ is of low-rank, and propose an MC algorithm based on subgradient descentnamed co-completion (COCO) motivated by elastic net and one-bit MC. We give atheoretical bound on the recovery effect of COCO and demonstrate its practicalusefulness through experiments.

Matrix Co-completion for Multi-label Classification with Missing Features and Labels