PP^T\Omega P P^T \eta_i + \epsilon_{y,i},  \quad \epsilon_{y,i} \sim N(0,\sigma^2) \nonumber,  \\ 
& X_i = \Lambda P P^T \eta_i+ \epsilon_i,   \quad  \epsilon_i \sim N_p(0,\Psi),\end{aligned}$$ where $P$ is a $k \times k$ orthogonal matrix $P$ ($PP^T =I$). However, the induced main effects satisfy $$\begin{aligned}
 \beta_X = \Psi^{-1} \Lambda  P ( P^T \Lambda^T  \Psi^{-1} \Lambda  P + P^T P)^{-1}  P^T \omega = \Psi^{-1} \Lambda ( \Lambda^T  \Psi^{-1} \Lambda  + I)^{-1} \omega.\end{aligned}$$ The same holds for induced interactions, showing that we do not need to impose identifiability constraints on $\Lambda$. We choose the Dirichlet-Laplace (DL) prior of [@Bhattacharya2015] row-wise, corresponding to $$\begin{gathered}
\lambda_{j,h} | \phi_{jh}, \tau_j \sim DE(\phi_{jh} \tau_j) \ \ \ \ \ \ \ h = 1, \cdots, k  \\
\phi_j \sim Dir(a,\cdots,a) \ \ \ \ \ \ \tau_j \sim Gamma(k a , 1/2),\end{gathered}$$ where $j = 1,\cdots,p$, $\phi_j = (\phi_{j1},\cdots,\phi_{jk})$, DE refers to the zero mean double-exponential or Laplace distribution, and $k$ is an upper bound on the number of factors, as the prior allows effective deletion of redundant factor loadings through row-wise shrinkage. The DL prior provides flexible shrinkage on the factor loadings matrix, generalizing the Bayesian Lasso [@park2008bayesian] to have a carefully chosen hierarchical structure on exposure-specific 