Robustness of Generalized Median Computation for Consensus Learning in Arbitrary Spaces

Robustness in terms of outliers is an important topic and has been formally studied for a variety of problems in machine learning and computer vision. Generalized median computation is a special instance of consensus learning and a common approach to finding prototypes. Related research can be found in numerous problem domains with a broad range of applications. So far, however, robustness of generalized median has only been studied in a few specific spaces. To our knowledge, there is no robustness characterization in a general setting, i.e. for arbitrary spaces. We address this open issue in our work. The breakdown point >=0.5 is proved for generalized median with metric distance functions in general. We also study the detailed behavior in case of outliers from different perspectives. In addition, we present robustness results for weighted generalized median computation and non-metric distance functions. Given the importance of robustness, our work contributes to closing a gap in the literature. The presented results have general impact and applicability, e.g. providing deeper understanding of generalized median computation and practical guidance to avoid non-robust computation.

Kernel-Based Generalized Median Computation for Consensus Learning

Computing a consensus object from a set of given objects is a core problem in machine learning and pattern recognition. One popular approach is to formulate it as an optimization problem using the generalized median. Previous methods like the Prototype and Distance-Preserving Embedding methods transform objects into a vector space, solve the generalized median problem in this space, and inversely transform back into the original space. Both of these methods have been successfully applied to a wide range of object domains, where the generalized median problem has inherent high computational complexity (typically $\mathcal{NP}$-hard) and therefore approximate solutions are required. Previously, explicit embedding methods were used in the computation, which often do not reflect the spatial relationship between objects exactly. In this work we introduce a kernel-based generalized median framework that is applicable to both positive definite and indefinite kernels. This framework computes the relationship between objects and its generalized median in kernel space, without the need of an explicit embedding. We show that the spatial relationship between objects is more accurately represented in kernel space than in an explicit vector space using easy-to-compute kernels, and demonstrate superior performance of generalized median computation on datasets of three different domains. A software toolbox resulting from our work is made publicly available to encourage other researchers to explore the generalized median computation and applications.