The LASSO algorithm gets "thrown off" when y is unscaled.
Thus, learn $\tilde{y}=\sigma_y^{-1}(y-\mu_y)$ instead when learning sparse coefficients $\tilde{\beta_j}$, and then add rescaled coefficients $\sigmay \sigma{x_j}^{-1} \tilde{\beta}j$ into $H{j}$ corresponding row of $y$.
The LASSO algorithm gets "thrown off" when
yis unscaled. Thus, learn $\tilde{y}=\sigma_y^{-1}(y-\mu_y)$ instead when learning sparse coefficients $\tilde{\beta_j}$, and then add rescaled coefficients $\sigmay \sigma{x_j}^{-1} \tilde{\beta}j$ into $H{j}$ corresponding row of $y$.