 For population recovery under the deletion channel , Ban et . al. provided an algorithm that could learn \ell-sparse distributions over strings using \exp %DIFDELCMD < \left(%%% \big( n^{1/2} \cdot (\log n)^{O(\ell)} %DIFDELCMD < \right) %%% \big) samples.
<clarity>  For population recovery under the deletion channel , Ban et . al. provided an algorithm that could learn \ell-sparse distributions over strings using \exp %DIFDELCMD < \left(%%% \big( n^{1/2} \cdot (\log n)^{O(\ell)} %DIFDELCMD < \right) %%% \big) samples.
 For population recovery under the deletion channel . Ban et al. showed how to learn \ell-sparse distributions over strings using \exp %DIFDELCMD < \left(%%% \big( n^{1/2} \cdot (\log n)^{O(\ell)} %DIFDELCMD < \right) %%% \big) samples.