Results
To understand the difficulty of the inversion problem,
we inverted the features using linear and quadratic programming
(LP and QP) solvers, which use different regularization measures.
In the LP solver, we solved for the positivevalued vector
that met (5.12), and minimized
(sum of the elements of ).
We imposed a lower bound of
to prevent
zerovalued elements.
Because is fixed by constraint (5.12),
the minimization had no effect, but the result provides us
at least with a valid starting point.
For the QP solver, we solved for the positivevalued vector
that met (5.12), and minimized
(sum of the squared elements of ).
Both solutions are shown in Figure 5.10
as well as the original raw spectrum.
The LP solution has only three spectral values that
exceed the lower bound! This solution might be useful
if sparseness is desired, but is clearly a bad spectral
estimate. The QP solution is more reasonable, but
reaches the lower bound for more than half of the spectrum.
Neither are good approximations to the original raw spectrum.
It is interesting to note that despite the wide differences
in how all these spectral estimates look,
all of the spectra are on the manifold
.
This illustrates the difficulty of feature inversion!
Clearly, meeting the constraint (5.12) is not enough.
See software/test_linexp.m.
Figure 5.10:
Linear and quadratic programming results.
Light jagged line: original raw spectrum. Circles: LP solution,
Dots: QP solution.

Sample spectra were generated using MCMCUMS
using the systematic approach.
We also computed the MaxEnt solution
by solving (5.15).
Results are shown in Figure 5.11.
The graph shows five separate spectra:
 Input spectrum : light jagged line,
 Typical MCMCUMS sample : dark jagged dashed line.
 Average of 100000 full MCMCUMS iterations (systematic approach): circles.
Each full iteration is simple iterations,
updating each of the free manifold dimensions.
 Autoregressive spectrum obtained by Levinson algorithm
(traditional AR spectral estimate) : triangles.
 MaxEnt solution
, solving (5.15) :
smooth dark solid line that runs through circles and triangles.
Once again, all of the spectra are on the manifold
^{5.1}.
The important takeaway from Figure 5.11
is that
runs directly through the
MCMCUMS sample mean values (circles),
confirming the reasoning in section 5.3.2
for approximating the asymptotic mean of MCMCUMS.
The fact that
also appears to be the same as
the traditional AR spectrum (aside for
a slight deviation at the end bins),
is not unexpected because it involves the same optimization as
for traditional MaxEnt spectral estimation, which is known
to coincide with the autoregressive method [34,35].
Despite the fact that
corresponds
to existing methods in this case, it is enlightening to
see a new interpretation in terms of sampling uniformly
(with maximum entropy) on the manifold.
It also leaves open the possibility that the sampling
viewpoint might be more general  something we confirm later.
See software/test_linexp.m.
Figure 5.11:
Combined spectral graph with five spectral estimates (see text for key).

Baggenstoss
20170519