vector is computed, and then the most correlated sample in the dataset is matched with the computed singular vector. Next, all data are projected onto the null space of the matched sample. The projected data are ready to perform one more iteration, if needed. These iterations are terminated either by reaching the desired number of selected samples or a given threshold of residual energy. Next, we present a lemma that guarantees the existence of a highly correlated sample with the first right singular vector.

\[lem:cor\] Let $\boldsymbol{a}_1, \boldsymbol{a}_2,\ldots, \boldsymbol{a}_M\in \mathbb{R}^N$ be $M$ given data points of dimension $N$. Let $\boldsymbol{A}$ denotes an $M\times N$ matrix with $\boldsymbol{a}_m^T$ being its $m^{th}$ row for $m=1,2,\ldots, M$. Let $\sigma_1$, $\boldsymbol{u}$ and $\boldsymbol{v}$ denote the first singular value, the corresponding left and right singular vectors of $\boldsymbol{A}$, respectively. Then, there exists at least one data point such that the absolute value of its inner product with $\boldsymbol{v}$ is greater than or equal to $\frac{\sigma_1}{\sqrt{M}}$. Hence, $$\small
\underset{m}{\text{max}}\;| \boldsymbol{v}^T {\boldsymbol{a}}_m|\geq \frac{\sigma_1}{\sqrt{M}}.$$

The following proposition states a lower bound on the maximum of the absolute value of the correlation between data points $\boldsymbol{a}_1, \boldsymbol{a}_2,\ldots, \boldsymbol{a}_M$ and $\boldsymbol{v}$, when data are normalized on the unit sphere. First, 