original NP-hard problem has been suggested in terms of the [D-optimal]{} and [A-optimal]{} solutions [@Boyd:2004:CO:993483; @Joshi2009Sensor_selection]. In addition to convex relaxation, a sub-modular cost function as the criterion of selection, allows us to employ much faster greedy optimization methods for selection [@sens_sel_greedy5717225]. The stochastic implementation of D-optimal solution is referred to *volume sampling* (VS), which is a fast and well-studied method. VS selects each subset of data, which are organized in the rows of a matrix, with probability proportional to the determinant (volume) of the reduced matrix. Moreover, QR decomposition with column pivoting (QRCP) and convex hull-based selection methods have been suggested for optimal data selection [@duersch2017randomized; @ding2018fast]. All the mentioned methods aim to select the most diverse subset of data in an optimal sense. However, these methods do not guarantee that the un-selected samples are well-covered by the selected ones. Further, outliers are selected with a high probability using such algorithms due to their diversity. Authors in [@joneidi2018Eoptimal] address this problem via a two-phase algorithm. There are some other efforts for outlier rejection in the selection procedure [@rahmani2017robust; @wang2017representative]. However, the outlier and inlier data are not well-defined and these methods are not consistent with general data.

There is 