Complete Generic Identifiability in Cyclic LiNGAM with General Confounding

Let $V=\{1,\dots,p\}$ index observed variables $X=(X_1,\dots,X_p)^\top\in\mathbb{R}^p$. Consider the linear structural equation model with feedback

equivalently $X_i=\sum_{j\in V\setminus\{i\}}\beta_{ij}X_j+U_i$ for each $i$, where $B=(\beta_{ij})_{i,j=1}^p$ has zero diagonal and $I-B^\top$ is invertible (so $X=(I-B^\top)^{-1}U$ is well defined). Directed cycles in the observed graph are allowed: define $G_{\mathrm{dir}}$ on $V$ by $j\to i$ iff $\beta_{ij}\neq 0$.

This setup follows Tramontano et al. (2025).

Assume the disturbance vector $U=(U_1,\dots,U_p)^\top$ is generated from latent exogenous noise as follows: there exist independent non-Gaussian random variables $\varepsilon_1,\dots,\varepsilon_m$ and measurable functions $g_i:\mathbb{R}^m\to\mathbb{R}$ such that

Hence the joint law of $U$ may have arbitrary dependence (including higher-order confounding) induced by shared latent exogenous sources and nonlinear mixing. The observational input is only the joint law $\mathcal{L}(X)$.

Fix a graphical model class $\mathcal{M}(\mathcal{G})$ specified by: which directed coefficients $\beta_{ij}$ are structurally allowed/nonzero (the directed part), and which latent confounding patterns are structurally allowed (e.g., via latent nodes pointing to subsets of observed nodes, equivalently a latent-projection/hyperedge specification). For a directed edge parameter $\beta_{ij}$, say $\beta_{ij}$ is generically identifiable in $\mathcal{M}(\mathcal{G})$ if there exists an exceptional set of parameter values of measure zero in the finite-dimensional structural parameters (at least the directed coefficients, and any finite-dimensional confounding-structure parameters if present) such that, outside that set, equality of observational laws implies equality of that coefficient: if $\mathcal{L}_\theta(X)=\mathcal{L}_{\theta'}(X)$ then $\beta_{ij}(\theta)=\beta_{ij}(\theta')$.