Identifiability of overcomplete independent component analysis

Independent component analysis (ICA) studies mixtures of independent latent sources. An ICA model is identifiable if the mixing can be recovered uniquely. It is well-known that ICA is identifiable if and only if at most one source is Gaussian. However, this applies only to the setting where the number of sources is at most the number of observations. In this paper, we generalize the identifiability of ICA to the overcomplete setting, where the number of sources exceeds the number of observations. We give an if and only if characterization of the identifiability of overcomplete ICA. The proof studies linear spaces of rank one symmetric matrices. For generic mixing, we present an identifiability condition in terms of the number of sources and the number of observations. We use our identifiability results to design an algorithm to recover the mixing matrix from data and apply it to synthetic data and two real datasets.