[@enet] can sometimes improve the performance of the lasso. The elastic net penalty is the weighted sum of the $\ell_{2}$ and $\ell_{1}$ norms of the coefficient vector to be estimated: $P_{\alpha}(\beta)=\frac{(1-\alpha)}{2}||\beta||_{2}^2 + \alpha||\beta||_{1}$. It is equivalent to the ridge regression penalty when $\alpha = 0$, and to the lasso penalty when $\alpha =1$. The elastic net solves the following problem:

$$\label{eq:en}
\hbeta = {\mathop{\mathrm{argmin}}}_{\beta\in\R^p} \, 
\half\|y-X\beta\|_2^2 + \lambda P_{\alpha}(\beta).$$

[The elastic net penalty is strictly convex, by strict convexity of the $\ell_{2}$ norm. Using this fact, the authors provide an upper bound on the distance between coefficients that correspond to highly correlated predictors.]{} This guarantees the grouping effect of the elastic net. Moreover, the elastic net solution can have more than $n$ non-zero coefficients, even when $p>n$, since it is equivalent to solving the lasso on an augmented dataset.

It is easy to see that the elastic net regularizes the feature covariance matrix from $X^TX$ to a form $X^TX+ \frac{1-\alpha}{2}\cdot\lambda I_p$ where $I_p$ is the $p\times p$ identity matrix. By inflating the diagonal it reduces the effective size of the off-diagonal correlations. [If the feature covariance matrix is block diagonal, its connected components correspond to groups of predictors that are 