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In the Lasso L1 regularization, from where comes the value of the variable $k$ in the second part of the function? Why isn't it $n$, too?

$$L(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T \cdot \beta)^2 + \lambda \sum_{j = 1}^k l(\beta_j)$$

$\beta$ is the parameter vector, $y$ the output vector and $x$ the input vector.

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$k$ is the length of $\beta$, the number of coefficients. The penalty is a function of the coefficients, not of the data.

Note, however, as a minor point, that you don't have to apply the penalty to all the coefficients in the model, although in your formulation you are.

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  • $\begingroup$ Right, I thought the dimension for instance $\beta$ and $y$ have to be the same, that's why I had the confusion. $\endgroup$
    – Mahoni
    May 7 '12 at 19:13
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    $\begingroup$ (+1) In particular, if you happen to not be using centered predictors and you do have an intercept in the model, you probably don't want to include it in the penalization. $\endgroup$
    – cardinal
    May 7 '12 at 20:05

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