  We study the property of the Fused Lasso Signal Approximator (FLSA) for
estimating a blocky signal sequence with additive noise. We transform the FLSA
to an ordinary Lasso problem. By studying the property of the design matrix in
the transformed Lasso problem, we find that the irrepresentable condition might
not hold, in which case we show that the FLSA might not be able to recover the
signal pattern. We then apply the newly developed preconditioning method --
Puffer Transformation [Jia and Rohe, 2012] on the transformed Lasso problem. We
call the new method the preconditioned fused Lasso and we give non-asymptotic
results for this method. Results show that when the signal jump strength
(signal difference between two neighboring groups) is big and the noise level
is small, our preconditioned fused Lasso estimator gives the correct pattern
with high probability. Theoretical results give insight on what controls the
signal pattern recovery ability -- it is the noise level {instead of} the
length of the sequence. Simulations confirm our theorems and show significant
improvement of the preconditioned fused Lasso estimator over the vanilla FLSA.
