Method of Sieves with Data Driven Basis Functions Consider a nonparametric regression problem with i.i.d. sampled data $(y_1,x_1), (y_2,x_2),\ldots, (y_n,x_n)$ and regression function
$$y_i = g_0(x_i) + \varepsilon_i,\quad \mathbb E[\varepsilon_i | x_i] = 0$$
One common approach to fitting $g_0$ non-parametrically is the so-called method of Sieves where for each $K$, we use some basis functions $p^K(x) = (p^{1K}(x),\ldots,p^{KK}(x))$ and we increase the complexity of the model $K$ as a function of $n$ so that
$\hat g(x) = (p^{K(n)})'\hat\beta_{K(n)}$ where $\hat\beta_{K(n)} = \underset{\beta}{\mathrm{argmin}} \sum_{i=1}^n (y_i - (p^K(x_i))'\beta)^2$.
I am in particular interested in a setting like Newey (1997), where we are interested in estimating $g_0$ for the purposes of computing some functional $\theta_0 = a(g_0)$ via a plug-in estimator $\hat \theta_n = a(\hat g_n)$. The paper is able to show asymptotic normality of such an estimator under certain conditions. In the paper, it is essentially assumed that the $p^K$ are fixed ex-ante. I am curious about a case where the $p^K$ are instead formed in a data-driven way and hence are allowed to be stochastic. It seems to me that the proof given in the paper could be modified to accommodate for $p^K$ being data driven, but I was wondering if there is already literature on a setting like this.

I have been thinking about similar questions in the functional regression context. Data driven orthogonal basis selection for functional data analysis is a paper I have found, though it's not for the setup in your question. (I wanted to comment but don't have enough reputation).
https://doi.org/10.1016/j.jmva.2021.104868
