# Significance of a source phase in a complex ICA

I am using complex-valued ICA to extract sources for complex-valued sensor data. One of the three ambiguities for complex ICA is phase ambiguity, i.e., phase rotation $\exp(i\theta_k)$ of the sources $s_k$ (in addition to permutation and scaling ambiguities, as for real ICA). Yet the complex linear ICA model $\bf{z}=A\cdot \bf{s}$ starts from the general form that all three quantities $\bf{z},A,\bf{s}$ are complex, so the merit of this assumption for $\bf{s}$ eludes me. Is there any meaning that can be given to the phases of the $s_k$, or is $|s_k|$ all that can be reported?

## 1 Answer

ICA is run on multiple samples: for a matrix $A$ and collections of vectors $\{s_n\}_{n=1}^N$ and $\{z_n\}_{n=1}^N$, the model says $z_n \approx As_n$. The $z$'s are observed, while $A$ and the $s$'s are inferred. If each entry of $s$ is allowed to be complex, then the algorithm may discover that $s_{1,1}$ and $s_{2, 1}$ are out of phase, meaning their ratio is not real. If entries of $s$ are all real, that situation cannot arise. So, it does make a difference; you can't just (for example) squeeze an extra diagonal matrix in there and make the phase difference of the latent variables go away.