Multi-label classification is a central problem in many application domains. In this paper, we present a novel supervised bi-directional model that learns a low-dimensional mid-level representation for multi-label classification. Unlike traditional multi-label learning methods which identify intermediate representations from either the input space or the output space but not both, the mid-level representation in our model has two complementary parts that capture intrinsic information of the input data and the output labels respectively under the autoencoder principle while augmenting each other for the target output label prediction. The resulting optimization problem can be solved efficiently using an iterative procedure with alternating steps, while closed-form solutions exist for one major step. Our experiments conducted on a variety of multi-label data sets demonstrate the efficacy of the proposed bi-directional representation learning model for multi-label classification.
We present tight upper and lower bounds on the spanning ratio of a large family of constrained θ-graphs. We show that constrained θ-graphs with 4k2 (k≥ 1 and integer) cones have a tight spanning ratio of 1+2 sin(θ/2), where θ is 2 π/ (4k+2). We also present improved upper bounds on the spanning ratio of the other families of constrained θ-graphs.