Conditions
for Linearity of Optimal
Estimation
The analog mapping problem is
strongly connected to the problem of
optimal estimation. In (asmyptotically
high delay) digital communications,
"structured" codes, such as linear
codes, lattice codes..., are
commonly used. Fortunately, these
structured codes can achieve the
fundamental limits, i.e., structure
comes free. In delay limited mappings,
this may not be the case. In this
work, we study the conditions
for which the optimal mappings,
specifically the estimators (decoders)
are structured, i.e., linear.
Specifically, we focus on the setting
below, looking for conditions for
h(Y)=kY for some k.
Our
main result is the necessary and
sufficient condition for linearity
of optimal estimation, in terms of
the characteristic function of the
source X, Fx(w) and the
noise Z, Fz(w), for a
given SNR level, snr: Fx(w)=(Fz(w))snr
. This connection, yields to several
important result. One of them is the
(asmyptotic) linearity of optimal
estimation at high snr, for a
Gaussian source, irrespective of the
channel. The numerical example below
demonstrates how the optimal
estimator converges to linear, for a
Gaussian source and uniform channel
(common setting in source
coding), as snr increases.
Related publications:
E. Akyol, K. Viswanatha and K. Rose, ``Linearity conditions for optimal
estimation from multiple noisy measurements", To appear in IEEE
Statistical Signal Processing Workshop (SSP), August, 2012.
E. Akyol, K. Viswanataha, and K.
Rose, On Conditions for Linearity of
Optimal Estimation, IEEE
Transactions on Information Theory,
to appear
E. Akyol, K. Viswanataha, and
K. Rose, On Conditions for Linearity
of Optimal Estimation, Proc. IEEE
Information Theory Workshop, Aug 2010
E. Akyol, K. Viswanataha, and K. Rose,
On Multidimensional Optimal
Estimators: Linearity Conditions,
Proc. IEEE Statistical Signal
Processing Workshop, June 2011
|