In a regression setting with input-output pairs $(x_n, y_n)$ for $n =1, . . . , N$, where the inputs $x_n = (x_{n,1}, . . . , x_{n,D})$ are generated by: $$x_{n,d} \sim N(0, s_d/N),$$
for dimension $d = 1, . . . , D$.
$X$ denotes the input matrix and $X^TX$ is a diagonal matrix with diagonal elements $(s_1, . . . , s_D)$.
How can you show that the estimated ridge weights simplify to:
$$\hat{w}_d^{Ridge} = \frac{s_d}{s_d + \lambda} \hat{w}_d^{LS}$$ for $d = 1, . . . , D$, where $\lambda$ denotes the ridge penalty parameter and $\hat{\mathbf{w}}^{LS}$ denotes the least squares estimates of the regression weights?
