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Binary segmentation, which is sequential in nature is thus far the mostwidely used method for identifying multiple change points in statisticalmodels. Here we propose a top down methodology called arbitrary segmentationthat proceeds in a conceptually reverse manner. We begin with an arbitrarysuperset of the parametric space of the change points, and locate unknownchange points by suitably filtering this space down. Critically, we reframe theproblem as that of variable selection in the change point parameters, thisenables the filtering down process to be achieved in a single step with the aidof an $\ell_0$ regularization, thus avoiding the sequentiality of binarysegmentation. We study this method under a high dimensional multiple changepoint linear regression model and show that rates convergence of the error inthe regression and change point estimates are near optimal. We propose asimulated annealing (SA) approach to implement a key finite state spacediscrete optimization that arises in our method. Theoretical results arenumerically supported via simulations. The proposed method is shown to possessthe ability to agnostically detect the `no change' scenario. Furthermore, itscomputational complexity is of order $O(Np^2)$+SA, where SA is the cost of a SAoptimization on a $N$(no. of change points) dimensional grid. Thus, theproposed methodology is significantly more computationally efficient thanexisting approaches. Finally, our theoretical results are obtained under weakermodel conditions than those assumed in the current literature.

Detection and estimation of parameters in high dimensional multiple  change point regression models via $\ell_1/\ell_0$ regularization and  discrete optimization