1Change-Point Estimation in High Dimensional Linear Regression Models via Sparse Group Lasso

We consider high dimensional linear regression models where p n 9 0 or p>> n, where p is the number of parameters in coefficients and n is the number of data sample. The coefficients in the linear regression model are sparse and has multiple change-points in n data samples. Our goal is to estimate the number and locations of change-points in the high dimensional linear regression model and estimate sparse coefficients in each of the intervals between change-points. This paper develops a approach solving multiple change-points estimation problem in high dimensional linear regression model based on sparse group Lasso (SGL). We analyze the performance of our approach and theoretical results about consistency are given. In particular, under certain assumptions and using a properly chosen regularization parameter, we show that estimation error of linear coefficients and change-point locations can be expressed as a function of n, p and s, where s is the sparsity level of each coefficient. Furthermore, we show that the isolated estimate change-points between true change-points occur with a small probability. Numerical simulations are provided to illustrate the effectiveness of our approach