Multiple kernel aggregation using fuzzy integrals

The so-called kernel-trick is a well-known method for mapping data in a lower dimensional space into a higher dimensional space to measure the similarity (inner product) of the data elements without ever explicitly performing the mapping. The hope is to induce an improved feature space in which to carry out pattern analysis. However, important questions remain, such as i) what is the best kernel, and ii) do some features or sensors require different kernels? One elegant way to address these problems is multiple kernel (MK) aggregation. To date, the research on MKs has predominately studied linear aggregation of kernels, namely weighted sums, e.g., conic and convex sums. In this paper, we propose a new method for kernel aggregation, fuzzy integral aggregation of MKs (FI-MK). We study different FI formulations to determine which ensures production of an aggregated kernel that is a valid Mercer kernel. We show that the Choquet integral (CI) achieves this goal for matrix-wise aggregation. We leverage our theoretical results to propose a genetic algorithm-based classification scheme called FIGA. Experiments on publicly available data sets are provided that demonstrate our FIGA algorithm produces superior results in the context of support vector machine (SVM)-based classification.