The generalized spike process, sparsity, and statistical independence, in
Modern Signal Processing (D. Rockmore and D. Healy, Jr., eds.), MSRI Publications, vol.46, pp.317-340, Cambridge University Press, 2004.
We consider the best sparsifying basis (BSB) and
the kurtosis maximizing basis (KMB) of a particularly
simple stochastic process called the "generalized spike process".
The BSB is a basis for which a given set of realizations of
a stochastic process can be represented most sparsely, whereas
the KMB is an approximation to the least statistically-dependent basis
(LSDB) for which the data representation has minimal statistical dependence.
In each realization, the generalized spike process puts a single spike with
amplitude sampled from the standard normal distribution at a random location
in an otherwise zero vector of length n.
We prove: 1) both the BSB and the KMB select the standard basis (if we restrict
our basis search to all possible orthonormal bases in Rn);
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
while the BSB seems insensitive.
Our results provide new additional support for the preference of the BSB over
the LSDB/KMB for data compression.
We include an explicit computation of the BSB for Meyer's discretized ramp process.
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