Residual sum of squares and Lasso regression

Question

Residual sum of squares (RSS) is defined by formula: $$RSS(\beta)=\sum_{i=1}^{n} (y_{i}-\sum_{j=1}^{p} \beta_{j}x_{ij})^2$$. In Lasso we minimize the RSS subject to $\sum_{j=1}^{p} |\beta_{j}| \leq s$.

Suppose that $p=2$, I don't see why $RSS$ is constant on elipses that are centered around $\hat\beta$ which is least square estimator. I don't even see why function $RSS(\beta)$ is ellipse for $\beta_{1}$ and $\beta_{2}$. I have tried the sum write in simple form but it didn't helped me.

Thanks for help.

Answer 1

For Lasso the constraint $\sum_{j=1}^p |\beta_j|\leq s$ is not an elipse. You are considering the betas for which $\|\beta\|_1\leq s$, i.e., in the case $p=2$, the points inside the square with vertices $\{(1,0), (0,1), (-1,0), (0,-1)\}$.