Identifiability of overcomplete independent component analysis
arxiv(2024)
摘要
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.
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