Estimating interactions from non-interacting features Suppose I have a sample $\mathcal{D}=\{(\mathbf{x}^{i}, y^{i})\}_{i=1\dots M}$ of binary variables $\mathbf{X}$ ($N$ of them) and a continuous variable $Y$ that I want to predict based on a linear regression with second order interactions:
$
\begin{align}
  y^{i} = \beta_{0} + \beta_{1}x^{i}_{1} + \dots \beta_{p}x^{i}_{N}x^{i}_{N-1}.
\end{align}
$
However, for each observation $i$ in the sample, we only observe one feature $k$ simultaneously. That is, for $x^{i}_{k}=1$ and $x^{i}_{j}=0$ for $j\neq k$.
Is there any way to estimate or give some bounds on the parameters of the interactions? It is ok to have a biased estimation, as long as the bias is more or less understood. I appreciate potential references too.

Unfortunately, no. You only observe the main effects. Your observations would be perfectly consistent with any interaction effect, small or large, positive or negative.
You can verify this yourself: take the data you have, with the main effect observations only, and then add a single "dummy" data row, with precisely one interaction between two of your features and any value of the target $y$ whatsoever. Then fit a model. You will see that your model will happily fit this completely random "observation" without any problem whatsoever, and importantly, without changing the parameter estimates on the main effect. This illustrates that your "main effect observations only" dataset has unfortunately zero predictive power for interactions.
