  In this paper, we compare 5 different nonlinear kernels: min-max, RBF, fRBF
(folded RBF), acos, and acos-$\chi^2$, on a wide range of publicly available
datasets. The proposed fRBF kernel performs very similarly to the RBF kernel.
Both RBF and fRBF kernels require an important tuning parameter ($\gamma$).
Interestingly, for a significant portion of the datasets, the min-max kernel
outperforms the best-tuned RBF/fRBF kernels. The acos kernel and acos-$\chi^2$
kernel also perform well in general and in some datasets achieve the best
accuracies.
  One crucial issue with the use of nonlinear kernels is the excessive
computational and memory cost. These days, one increasingly popular strategy is
to linearize the kernels through various randomization algorithms. In our
study, the randomization method for the min-max kernel demonstrates excellent
performance compared to the randomization methods for other types of nonlinear
kernels, measured in terms of the number of nonzero terms in the transformed
dataset.
  Our study provides evidence for supporting the use of the min-max kernel and
the corresponding randomized linearization method (i.e., the so-called "0-bit
CWS"). Furthermore, the results motivate at least two directions for future
research: (i) To develop new (and linearizable) nonlinear kernels for better
accuracies; and (ii) To develop better linearization algorithms for improving
the current linearization methods for the RBF kernel, the acos kernel, and the
acos-$\chi^2$ kernel. One attempt is to combine the min-max kernel with the
acos kernel or the acos-$\chi^2$ kernel. The advantages of these two new and
tuning-free nonlinear kernels are demonstrated vias our extensive experiments.
