Convex Transduction with the Normalized Cut

We discuss approaches to transduction based on graph cut cost functions. More specifically, we focus on the normalized cut, which is the cost function of choice in many clustering applications, notably in image segmentation. Since optimizing the normalized cut cost is an NP-complete problem, much of the research attention so far has gone to relaxing the problem of normalized cut clustering to tractable problems, producing so far a spectral relaxation and a more recently a tighter but computationally much tougher semi-definite programming (SDP) relaxation. In this paper we deliver two main contributions: first, we show how an alternative SDP relaxation yields a much more tractable optimization problem, and we show how scalability and speed can further be increased by making a principled approximation. Second, we show how it is possible to efficiently optimize the normalized cut cost in a transduction setting using our newly proposed approaches. Positive empirical results are reported.