  The reduced-rank method exploits the distortion-variance tradeoff to yield
superior solutions for classic problems in statistical signal processing such
as parameter estimation and filtering. The central idea is to reduce the
variance of the solution at the expense of introducing a little distortion. In
the context of parameter estimation, this yields an estimator whose sum of
distortion plus variance is smaller than the variance of its undistorted
counterpart. The method intrinsically results in an ordering mechanism for the
singular vectors of the system matrix in the measurement model used for
estimating the parameter of interest. According to this ordering rule, only a
few \emph{dominant} singular vectors need to be selected to construct the
reduced-rank solution while the rest can be discarded. The reduced-rank
estimator is less sensitive to measurement errors. In this paper, we attempt to
derive the reduced-rank estimator for the total least squares (TLS) problem,
including the order selection rule. It will be shown that, due to the inherent
structure of the problem, it is not possible to exploit the distortion-variance
tradeoff in TLS formulations using existing techniques, except in some special
cases. This result has not been reported previously and warrants a new
viewpoint to achieve rank reduction for the TLS estimation problem. The work is
motivated by the problems arising in practical applications such as channel
estimation in wireless communication systems and time synchronization in
wireless sensor networks.
