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I am reading the paper about group lasso. The equation (3) in the paper has a number, $\sqrt p_{l}$, between left double bar of norm sign as: $\vert\sqrt p_{l}\vert\beta^{(l)}\vert\vert_{2}$ (where $\vert\vert \cdot \vert\vert$ is norm symbol). What is the operation here?

Excerpt from paper

The $l1$ norm penalty promotes sparsity in the solution vector $β$. Suppose, further, that our predictor variables were divided into $m$ different groups. We are given these group memberships and rather than sparsity in individual elements of $β, we would like a solution which uses only a few of the groups. Yuan and Lin [2007] proposed the Group Lasso criterion for this problem; to find

$min_{\beta} \frac{1}{2} \vert\vert y-\sum_{l=1}^m X^{(l)}\beta^{(l)}\vert\vert^2_{2} + \vert\sqrt p_{l}\vert\beta^{(l)}\vert\vert_{2} \tag{3}$

where $X_{(l)}$ is the submatrix of X with columns corresponding to the predictors in group $l$, $β_{(l)}$ is the coefficient vector of that group, and $W_{l}$ is some penalty matrix, $p_{l}$ is the number of covariates in group {(l)}

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This is most likely a typo for "$\sqrt{p_l} ||\beta^{(l)}||_2$". Compare with equations 7, 8, and 10.

Note that equation 3 has "$p_\ell$" instead of "$p_l$" and the only other odd use of "$\ell$" in this paper is "$W_\ell$" two lines before equation 3. This makes me think there could be a cut-and-paste splice here and not all minor variations in notation were caught.

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