  This paper studies the recovery of a joint piece-wise linear trend from a
time series using L1 regularization approach, called L1 trend filtering (Kim,
Koh and Boyd, 2009). We provide some sufficient conditions under which a L1
trend filter can be well-behaved in terms of mean estimation and change point
detection. The result is two-fold: for the mean estimation, an almost optimal
consistent rate is obtained; for the change point detection, the slope change
in direction can be recovered in a high probability. In addition, we show that
the weak irrepresentable condition, a necessary condition for LASSO model to be
sign consistent (Zhao and Yu, 2006), is not necessary for the consistent change
point detection. The performance of the L1 trend filter is evaluated by some
finite sample simulations studies.
