Sobolev functions approximated by ridge functions

Let $f \in W^{k,2}(\mathbb{R}^d)$, a Sobolev space with smoothness $k$ and dimension $d$. We aim to approximate$f$ using ridge functions of the form $g(\mathbf{a}.\mathbf{x})$. Suppose the approximation error after using $n$ ridge functions is denoted by $E_n(f)$. Can we construct a class of $f$ having a certain smoothness, that the error $E_n(f)$ when approximated using ridge functions satisfies the exponential decay?$$E_n(f) \le C\exp(-\alpha n^{\frac{1}{d}})$$