Non-Independent Components Analysis

A seminal result in the ICA literature states that for $AY = \varepsilon$, if the components of $\varepsilon$ are independent and at most one is Gaussian, then $A$ is identified up to sign and permutation of its rows [Comon, 1994]. In this paper we study to which extent the independence assumption can be relaxed by replacing it with restrictions on the cumulants of $\varepsilon$. We document minimal cumulant conditions for identifiability and propose efficient estimation methods based on the new identification results. In situations where independence cannot be assumed the efficiency gains can be significant relative to methods that rely on independence. The proof strategy employed highlights new geometric and combinatorial tools that can be adopted to study identifiability via higher order restrictions in linear systems.