  We develop the "generalized consistent weighted sampling" (GCWS) for hashing
the "powered-GMM" (pGMM) kernel (with a tuning parameter $p$). It turns out
that GCWS provides a numerically stable scheme for applying power
transformation on the original data, regardless of the magnitude of $p$ and the
data. The power transformation is often effective for boosting the performance,
in many cases considerably so. We feed the hashed data to neural networks on a
variety of public classification datasets and name our method ``GCWSNet''. Our
extensive experiments show that GCWSNet often improves the classification
accuracy. Furthermore, it is evident from the experiments that GCWSNet
converges substantially faster. In fact, GCWS often reaches a reasonable
accuracy with merely (less than) one epoch of the training process. This
property is much desired because many applications, such as advertisement
click-through rate (CTR) prediction models, or data streams (i.e., data seen
only once), often train just one epoch. Another beneficial side effect is that
the computations of the first layer of the neural networks become additions
instead of multiplications because the input data become binary (and highly
sparse).
  Empirical comparisons with (normalized) random Fourier features (NRFF) are
provided. We also propose to reduce the model size of GCWSNet by count-sketch
and develop the theory for analyzing the impact of using count-sketch on the
accuracy of GCWS. Our analysis shows that an ``8-bit'' strategy should work
well in that we can always apply an 8-bit count-sketch hashing on the output of
GCWS hashing without hurting the accuracy much. There are many other ways to
take advantage of GCWS when training deep neural networks. For example, one can
apply GCWS on the outputs of the last layer to boost the accuracy of trained
deep neural networks.
