  Binary segmentation, which is sequential in nature is thus far the most
widely used method for identifying multiple change points in statistical
models. Here we propose a top down methodology called arbitrary segmentation
that proceeds in a conceptually reverse manner. We begin with an arbitrary
superset of the parametric space of the change points, and locate unknown
change points by suitably filtering this space down. Critically, we reframe the
problem as that of variable selection in the change point parameters, this
enables the filtering down process to be achieved in a single step with the aid
of an $\ell_0$ regularization, thus avoiding the sequentiality of binary
segmentation. We study this method under a high dimensional multiple change
point linear regression model and show that rates convergence of the error in
the regression and change point estimates are near optimal. We propose a
simulated annealing (SA) approach to implement a key finite state space
discrete optimization that arises in our method. Theoretical results are
numerically supported via simulations. The proposed method is shown to possess
the ability to agnostically detect the `no change' scenario. Furthermore, its
computational complexity is of order $O(Np^2)$+SA, where SA is the cost of a SA
optimization on a $N$(no. of change points) dimensional grid. Thus, the
proposed methodology is significantly more computationally efficient than
existing approaches. Finally, our theoretical results are obtained under weaker
model conditions than those assumed in the current literature.
