$\mathbb{E}_\nu\{\|\boldsymbol{w}-\hat{\boldsymbol{w}} \|_2\}$, where $\mathbb{E}_\nu$ is expectation w.r.t. noise distribution of $\boldsymbol{w}-\hat{\boldsymbol{w}}$. This criterion is referred as *A-optimal* design in the literature of optimization. It is an NP hard problem, which can be solved via convex relaxation with computational complexity of $O(M^3)$ [@Joshi2009Sensor_selection].

However, there are other criteria which have interesting properties. For example *D-optimal* design optimizes the determinant of a reduced matrix [@Joshi2009Sensor_selection]. There are several other efforts in this area [@deshpande2010efficient; @deshpande2006matrix; @li2017polynomial; @farahat2015greedy]. Inspired by D-optimal design, volume sampling (VS), which has received lots of attention, considers a selection probability for each subset of data, which is proportional to the determinant (volume) of the reduced matrix [@li2017polynomial; @nikolov2018proportional; @Derezinski2018SubsamplingSampling]. VS theory expresses that if

Representative Selection
------------------------

A method for sampling from a set of data is proposed by Elhamifar et. al. based on sparse modeling representative selection (SMRS) [@Elhamifar2012SeeObjects]. Their proposed cost function for data selection is the error of projecting all the data onto the subspace spanned by the selected data. Mathematically, the optimization problem in [@Elhamifar2012SeeObjects] can be written as, However, there is no guarantee that convex relaxation provides the best approximation for an NP-hard problem. In this paper, we propose a new fast algorithm 