Manifold regularization in structured output space for semi-supervised structured output prediction

Structured output prediction aims to learn a predictor to predict a structured output from a input data vector. The structured outputs include vector, tree, sequence, etc. We usually assume that we have a training set of input-output pairs to train the predictor. However, in many real-world applications, it is difficult to obtain the output for a input, and thus for many training input data points, the structured outputs are missing. In this paper, we discuss how to learn from a training set composed of some input-output pairs and some input data points without outputs. This problem is called semi-supervised structured output prediction. We propose a novel method for this problem by constructing a nearest neighbor graph from the input space to present the manifold structure and use it to regularize the structured output space directly. We define a slack structured output for each training data point and propose to predict it by learning a structured output predictor. The learning of both slack structured outputs and the predictor are unified within one single minimization problem. In this problem, we propose to minimize the structured loss between the slack structured outputs of neighboring data points and the prediction error measured by the structured loss. The problem is optimized by an iterative algorithm. Experiment results over three benchmark data sets show its advantage.