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The Squared-Error of Generalized LASSO: A Precise Analysis
We consider the problem of estimating an unknown signal from noisy
linear observations . In many practical instances,
has a certain structure that can be captured by a structure inducing convex
function . For example, norm can be used to encourage a
sparse solution. To estimate with the aid of , we consider the
well-known LASSO method and provide sharp characterization of its performance.
We assume the entries of the measurement matrix and the noise vector
have zero-mean normal distributions with variances and
respectively. For the LASSO estimator , we attempt to calculate the
Normalized Square Error (NSE) defined as as
a function of the noise level , the number of observations and the
structure of the signal. We show that, the structure of the signal and
choice of the function enter the error formulae through the summary
parameters and , which are defined as the Gaussian
squared-distances to the subdifferential cone and to the -scaled
subdifferential, respectively. The first LASSO estimator assumes a-priori
knowledge of and is given by . We prove that its worst case NSE is achieved when
and concentrates around .
Secondly, we consider , for some
. This time the NSE formula depends on the choice of
and is given by . We then establish a mapping
between this and the third estimator . Finally, for a number of important structured signal classes,
we translate our abstract formulae to closed-form upper bounds on the NSE