Problem(7)isNP-complete. Weshow that there exists apolynomial time reduction from the set cover problem to(7). We construct theM matrix according to the sets A1,...,Am (thei-thcolumnof M isthenonzeropatternof Ai).

Whenlabelsaresemantically complex and multi-modal, it is more natural to assign a label to multiple semantic clusters. In product categorization, for instance, the tag "belt" can be related to a vehicle belt (under "vehicle accessories" category),oraman'sbelt(under "clothing" category).

Multi-label active learning is a crucial yet challenging area in contemporary machine learning, often complicated by a large and sparse label space.

See the "Ranker" block of Figure 1.

For instance, the label "Apple" can be the fruit or the brand name, which

Extreme multi-label classification aims to learn a classifier that annotates an instance with a relevant subset of labels from an extremely large label set. Many existing solutions embed the label matrix to a low-dimensional linear subspace, or examine the relevance of a test instance to every label via a linear scan. In practice, however, those approaches can be computationally exorbitant. To alleviate this drawback, we propose a Block-wise Partitioning (BP) pretreatment that divides all instances into disjoint clusters, to each of which the most frequently tagged label subset is attached. One multi-label classifier is trained on one pair of instance and label clusters, and the label set of a test instance is predicted by first delivering it to the most appropriate instance cluster. Experiments on benchmark multi-label data sets reveal that BP pretreatment significantly reduces prediction time, and retains almost the same level of prediction accuracy.