for solving Problem (\[eq:orig\_sel\]). Dissimilarity-based Sparse Subset Selection (DS3) algorithm selects a subset of data based on pairwise distance of all data to some target points [@Elhamifar2016DissimilaritySelection]. DS3 considers a source dataset and its goal is to encode the target data according to pairwise dissimilarity between each sample of source and target datasets. This algorithm can be interpreted as the non-linear implementation of SMRS algorithm [@Elhamifar2016DissimilaritySelection].

Iterative Projection and Matching (IPM) {#sec:ipm}
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In this section, an iterative and computationally efficient algorithm is proposed for approximating the solution to the NP-hard selection problem (\[eq:orig\_sel\]). The proposed algorithm iteratively finds the best direction on the unit sphere[^2], and then from the available samples in dataset selects the sample with the smallest angle to the found direction.

Projection of all the data on to the subspace spanned by the $K$ rows of $\boldsymbol{A}$, indexed by $\mathbb{T}$, i.e., $\pi_{\mathbb{T}}(\boldsymbol{A})$, can be expressed by a rank-$K$ factorization, $\boldsymbol{U}\boldsymbol{V}^T$, where $\boldsymbol{U}\in \mathbb{R}^{M\times K}$, $\boldsymbol{V}^T\in \mathbb{R}^{K\times N}$, and $\boldsymbol{V}^T$ includes the $K$ rows of $\boldsymbol{A}$, indexed by $\mathbb{T}$, and normalized to have unit length. Therefore, optimization problem (\[eq:orig\_sel\]) can be restated as $$\small 
\label{eq:rewrite}
\underset{\boldsymbol{U},\boldsymbol{V}}{\text{argmin}}\;\|\boldsymbol{A}-\boldsymbol{UV}^T\|_F^2 \;\small{\text{s.t.}}\; \boldsymbol{v}_k\in \mathbb{A},$$ where, $\mathbb{A}=\{\Tilde{\boldsymbol{a}_1},\Tilde{\boldsymbol{a}_2},\ldots, \Tilde{\boldsymbol{a}}_M\}$, $\Tilde{\boldsymbol{a}}_m=\boldsymbol{a}_m/\|\boldsymbol{a}_m\|_2$, and $\boldsymbol{v}_k$ is the 