quality:corruption, information:filtering
Orthonormal covariates

It can also be compared to regression with best subset selection, in which the goal is to minimize min β ∈ R p { 1 N ‖ y − X β ‖ 2 2 + λ ‖ β ‖ 0 } {\displaystyle \ \min _{\beta \in \mathbb {R} ^{p}}\left\{~{\tfrac {1}{N}}{\Bigl \|}\ y-X\beta \ {\Bigr \|}_{2}^{2}\ +\ \lambda \ {\Bigl \|}\ \beta \ {\Bigr \|}_{0}~\right\}\ } where ‖ ⋅ ‖ 0 {\displaystyle \ \|\cdot \|_{0}\ } is the " ℓ 0 {\displaystyle \ \ell ^{0}\ } norm", which is defined as ‖ z ‖ = m {\displaystyle \ \|z\|=m\ } if exactly m components of z are nonzero. Again assuming orthonormal covariates, it can be shown that in this special case β ^ j = H N λ (β ^ j OLS) = β ^ j OLS ⋅ I {\displaystyle \ {\hat {\beta }}_{j}\ =\ H_{\sqrt {N\lambda \ }}\ \left(\ {\hat {\beta }}{}_{j}^{\!\;{\text{OLS}}}\ \right)\ =\ {\hat {\beta }}{}_{j}^{\!\;{\text{OLS}}}\cdot \operatorname {\mathbb {I} } \left\ } where H α {\displaystyle \ H_{\alpha }\ } is again the hard thresholding operator and I {\displaystyle \ \mathbb {I} \ } is an indicator function (it is 1 if its argument is true and 0 otherwise).