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We consider the problem of learning classifiers for structurally incomplete data, where some objects have a subset of features inherently absent due to complex relationships between the features. The common approach for handling missing features is to begin with a preprocessing phase that completes the missing features, and then use a standard classification procedure. In this paper we show how incomplete data can be classified directly without any completion of the missing features using a max-margin learning framework. We formulate this task using a geometrically-inspired objective function, and discuss two optimization approaches: The linearly separable case is written as a set of convex feasibility problems, and the non-separable case has a non-convex objective that we optimize iteratively. By avoiding the pre-processing phase in which the data is completed, these approaches offer considerable computational savings. More importantly, we show that by elegantly handling complex patterns of missing values, our approach is both competitive with other methods when the values are missing at random and outperforms them when the missing values have non-trivial structure. We demonstrate our results on two real-world problems: edge prediction in metabolic pathways, and automobile detection in natural images.