  A basis under which a given set of realizations of a stochastic process can
be represented most sparsely (the so-called best sparsifying basis (BSB)) and
the one under which such a set becomes as less statistically dependent as
possible (the so-called least statistically-dependent basis (LSDB)) are
important for data compression and have generated interests among computational
neuroscientists as well as applied mathematicians. Here we consider these bases
for a particularly simple stochastic process called ``generalized spike
process'', which puts a single spike--whose amplitude is sampled from the
standard normal distribution--at a random location in the zero vector of length
$\ndim$ for each realization.
  Unlike the ``simple spike process'' which we dealt with in our previous paper
and whose amplitude is constant, we need to consider the kurtosis-maximizing
basis (KMB) instead of the LSDB due to the difficulty of evaluating
differential entropy and mutual information of the generalized spike process.
By computing the marginal densities and moments, we prove that: 1) the BSB and
the KMB selects the standard basis if we restrict our basis search within all
possible orthonormal bases in ${\mathbb R}^n$; 2) if we extend our basis search
to all possible volume-preserving invertible linear transformations, then the
BSB exists and is again the standard basis whereas the KMB does not exist.
Thus, the KMB is rather sensitive to the orthonormality of the transformations
under consideration whereas the BSB is insensitive to that. Our results once
again support the preference of the BSB over the LSDB/KMB for data compression
applications as our previous work did.
