## Multi-label Learning via Structured Decomposition and Group Sparsity

This paper is now available on arxiv：

http://arxiv.org/abs/1103.0102

In multi-label learning, each sample is associated with several labels. Existing works indicate that exploring correlations between labels improve the prediction performance. However, embedding the label correlations into the training process significantly increases the problem size. Moreover, the mapping of the label structure in the feature space is not clear.

In this paper, we propose a novel multi-label learning method “Structured Decomposition + Group Sparsity (SDGS)”. In SDGS, we learn a feature subspace for each label from the structured decomposition of the training data, and predict the labels of a new sample from its group sparse representation on the multi-subspace obtained from the structured decomposition.

In particular, in the training stage, we decompose the data matrix $X\in R^{n\times p}$ as $X=\sum_{i=1}^kL^i+S$, wherein the rows of $L^i$ associated with samples that belong to label $i$ are nonzero and consist a low-rank matrix, while the other rows are all-zeros, the residual $S$ is a sparse matrix. The row space of $L_i$ is the feature subspace corresponding to label $i$. This decomposition can be efficiently obtained via randomized optimization. In the prediction stage, we estimate the group sparse representation of a new sample on the multi-subspace via group \emph{lasso}. The nonzero representation coefficients tend to concentrate on the subspaces of labels that the sample belongs to, and thus an effective prediction can be obtained.

SDGS finds the mapping of labels in the feature space, where the label correlations are naturally preserved in the corresponding mappings. Thus it explores the label structure without increasing the problem size. SDGS is robust to the imbalance between positive and negative samples, because it uses group sparsity in the multi-subspace to select the labels, which considers both the discriminative and relative information between the mappings of labels in feature subspace.