Generalized Intersection Kernel

Following the very recent line of work on the “generalized min-max” (GMM) kernel [7], this study proposes the “generalized intersection” (GInt) kernel and the related “normalized generalized min-max” (NGMM) kernel. In computer vision, the (histogram) intersection kernel has been popular, and the GInt kernel generalizes it to data which can have both negative and positive entries. Through an extensive empirical classification study on 40 datasets from the UCI repository, we are able to show that this (tuning-free) GInt kernel performs fairly well. The empirical results also demonstrate that the NGMM kernel typically outperforms the GInt kernel. Interestingly, the NGMM kernel has another interpretation — it is the “asymmetrically transformed” version of the GInt kernel, based on the idea of “asymmetric hashing” [14]. Just like the GMM kernel, the NGMM kernel can be efficiently linearized through (e.g.,) generalized consistent weighted sampling (GCWS), as empirically validated in our study. Owing to the discrete nature of hashed values, it also provides a scheme for approximate near neighbor search. The proposed GInt kernel and NGMM kernel provide additional options for practitioners to deal with data in their specific domains. We also note that there have been a series of (unpublished) works on related topics. For example, [7] proposed the GMM kernel and the “normalized random Fourier features (NRFF)” method; [8] compared the Nystrom method for approximating the GMM kernel with NRFF; [10] developed some basic mathematical theory (e.g., the convergence and asymptotic normality) for the GMM kernel; [6] compared the hashed (linearized) GMM kernel with a series of kernels which can be approximated by sign stable random projections.