  The recently proposed "generalized min-max" (GMM) kernel can be efficiently
linearized, with direct applications in large-scale statistical learning and
fast near neighbor search. The linearized GMM kernel was extensively compared
in with linearized radial basis function (RBF) kernel. On a large number of
classification tasks, the tuning-free GMM kernel performs (surprisingly) well
compared to the best-tuned RBF kernel. Nevertheless, one would naturally expect
that the GMM kernel ought to be further improved if we introduce tuning
parameters.
  In this paper, we study three simple constructions of tunable GMM kernels:
(i) the exponentiated-GMM (or eGMM) kernel, (ii) the powered-GMM (or pGMM)
kernel, and (iii) the exponentiated-powered-GMM (epGMM) kernel. The pGMM kernel
can still be efficiently linearized by modifying the original hashing procedure
for the GMM kernel. On about 60 publicly available classification datasets, we
verify that the proposed tunable GMM kernels typically improve over the
original GMM kernel. On some datasets, the improvements can be astonishingly
significant.
  For example, on 11 popular datasets which were used for testing deep learning
algorithms and tree methods, our experiments show that the proposed tunable GMM
kernels are strong competitors to trees and deep nets. The previous studies
developed tree methods including "abc-robust-logitboost" and demonstrated the
excellent performance on those 11 datasets (and other datasets), by
establishing the second-order tree-split formula and new derivatives for
multi-class logistic loss. Compared to tree methods like
"abc-robust-logitboost" (which are slow and need substantial model sizes), the
tunable GMM kernels produce largely comparable results.
